Integrals

Q61 If $\int_0^a {\frac{1}{{1 + 4{x^2}}}} dx = \frac{\pi }{8},$ then $a =$.......... FillBlank Q62 $\int {\frac{{\sin x}}{{3 + 4{{\cos }^2}x}}} dx =$.......... FillBlank Q63 The value of $\int_{ - \pi }^\pi {{{\sin }^3}} x{\cos ^2}xdx$ is………. FillBlank Q35 $\int {\frac{{{x^2}}}{{{x^4} - {x^2} - 12}}} dx$ LA Q36 $\int {\frac{{{x^2}}}{{\left( {{x^2} + {a^2}} \right)\left( {{x^2} + {b^2}} \right)}}} dx$ LA Q37 $\int_0^\pi {\frac{x}{{1 + \sin x}}}$ LA Q38 $\int {\frac{{2x - 1}}{{(x - 1)(x + 2)(x - 3)}}} dx$ LA Q39 $\int {{e^{{{\tan }^{ - 1}}x}}} \left( {\frac{{1 + x + {x^2}}}{{1 + {x^2}}}} \right)dx$ LA Q40 $\int {{{\sin }^{ - 1}}} \sqrt {\frac{x}{{a + x}}} dx$ LA Q41 $\int_{\pi /3}^{\pi /2} {\frac{{\sqrt {1 + \cos x} }}{{{{(1 - \cos x)}^{5/2}}}}} dx$ LA Q42 $\int {{e^{ - 3x}}} {\cos ^3}xdx$ LA Q43 $\int {\sqrt {\tan x} } dx$ LA Q44 $\int_0^{\pi /2} {\frac{{dx}}{{{{\left( {{a^2}{{\cos }^2}x + {b^2}{{\sin }^2}x} \right)}^2}}}}$ LA Q45 $\int_0^1 x \log (1 + 2x)dx$ LA Q46 $\int_0^\pi x \log \sin xdx$ LA Q47 $\int_{\pi /4}^{\pi /4} {\log } (\sin x + \cos x)dx$ LA Q1 $\int {\frac{{2x - 1}}{{2x + 3}}} dx = x - \log \left| {{{(2x + 3)}^2}} \right| + C$

SA Q2 $\int {\frac{{2x + 3}}{{{x^2} + 3x}}} dx = \log \left| {{x^2} + 3x} \right| + C$

SA Q3 $\int {\frac{{\left( {{x^2} + 2} \right)d}}{{x + 1}}} x$

SA Q4 $\int {\frac{{{e^{6\log x}} - {e^{5\log x}}}}{{{e^{4\log x}} - {e^{3\log x}}}}} dx$

SA Q5 $\int {\frac{{(1 + \cos x)}}{{x + \sin x}}} dx$

SA Q6 $\int {\frac{{dx}}{{1 + \cos x}}}$

SA Q7 $\int {{{\tan }^2}} x{\sec ^4}xdx$ SA Q8 $\int {\frac{{\sin x + \cos x}}{{\sqrt {1 + \sin 2x} }}} dx$

SA Q9 $\int {\sqrt {1 + \sin x} } dx$

SA Q10 $\int {\frac{x}{{\sqrt x + 1}}} dx$

SA Q11 $\int {\sqrt {\frac{{a + x}}{{a - x}}} } dx$

SA Q12 $\int {\frac{{{x^{1/2}}}}{{1 + {x^{3/4}}}}} dx$ SA Q13 $\int {\frac{{\sqrt {1 + {x^2}} }}{{{x^4}}}} dx$ SA Q14 $\int {\frac{{dx}}{{\sqrt {16 - 9{x^2}} }}}$ SA Q15 $\int {\frac{{dt}}{{\sqrt {3t - 2{t^2}} }}}$ SA Q16 $\int {\frac{{3x - 1}}{{\sqrt {{x^2} + 9} }}} dx$ SA Q17 $\int {\sqrt {5 - 2x + {x^2}} } dx$ SA Q18 $\int {\frac{x}{{{x^4} - 1}}} dx$ SA Q19 $\int {\frac{{{x^2}}}{{1 - {x^4}}}} dx$ SA Q20 $\int {\sqrt {2ax - {x^2}} } dx$ SA Q21 $\int {\frac{{{{\sin }^{ - 1}}x}}{{{{\left( {1 - {x^2}} \right)}^{3/4}}}}} dx$ SA Q22 $\int {\frac{{(\cos 5x + \cos 4x)}}{{1 - 2\cos 3x}}} dx$ SA Q23 $\int {\frac{{{{\sin }^6}x + {{\cos }^6}x}}{{{{\sin }^2}x{{\cos }^2}x}}} dx$
SA Q24 $\int {\frac{{\sqrt x }}{{\sqrt {{a^3} - {x^3}} }}} dx$ SA Q25 $\int {\frac{{\cos x - \cos 2x}}{{1 - \cos x}}} dx$ SA Q26 $\int {\frac{{dx}}{{x\sqrt {{x^4} - 1} }}}$ SA Q27 $\int_0^2 {\left( {{x^2} + 3} \right)} dx$ SA Q28 $\int_0^2 {{e^x}} dx$ SA Q29 $\int_0^1 {\frac{{dx}}{{{e^x} + {e^{ - x}}}}}$ SA Q30 $\int_0^{\pi /2} {\frac{{\tan x}}{{1 + {m^2}{{\tan }^2}x}}} dx$ SA Q31 $\int_1^2 {\frac{{dx}}{{\sqrt {(x - 1)(2 - x)} }}}$ SA Q32 $\int_0^1 {\frac{x}{{\sqrt {1 + {x^2}} }}} dx$ SA Q33 $\int_0^\pi x \sin x{\cos ^2}xdx$ SA Q34 $\int_0^{1/2} {\frac{{dx}}{{\left( {1 + {x^2}} \right)\sqrt {1 - {x^2}} }}}$ SA

Q1 $\sin 2x$ SA Q2 $\cos 3x$ SA Q3 ${e^{2x}}$ SA Q4 ${\left( {ax + b} \right)^2}$ SA Q5 $\sin 2x - 4{e^{3x}}$ SA Q6 $\int {\left( {4{e^{3x}} + 1} \right)dx}$ SA Q7 $\int {{x^2}\left( {1 - \cfrac{1}{{{x^2}}}} \right)dx}$ SA Q8 $\int {\left( {a{x^2} + bx + c} \right)dx}$ SA Q10 $\int {{{\left( {\sqrt x - \cfrac{1}{{\sqrt x }}} \right)}^2}dx}$ SA Q11 $\int {\cfrac{{{x^3} + 5{x^2} - 4}}{{{x^2}}}dx}$ SA Q12 $\int {\cfrac{{{x^3} + 3x + 4}}{{\sqrt x }}} dx$ SA Q13 $\int {\cfrac{{{x^3} - {x^2} + x - 1}}{{x - 1}}dx}$ SA Q14 . $\int {\left( {1 - x} \right)\sqrt x dx}$ SA Q15 $\int {\sqrt x \left( {3{x^2} + 2x + 3} \right)dx}$ SA Q16 $\int {\left( {2x - 3\cos x + {e^x}} \right)dx}$ SA Q17 $\int {\left( {2{x^2} - 3\sin x + 5\sqrt x } \right)dx}$ SA Q18 $\int {\sec x\left( {\sec x + \tan x} \right)dx}$ SA Q19 $\int {\cfrac{{{{\sec }^2}x}}{{\cos e{c^2}x}}dx}$ SA Q20 $\int {\cfrac{{2 - 3\sin x}}{{{{\cos }^2}x}}dx}$ SA Q21 The anti derivative of $\left( {\sqrt x + \cfrac{1}{{\sqrt x }}} \right)$ equals

SA Q22 If $\cfrac{d}{{dx}}f\left( x \right) = 4{x^2} - \cfrac{3}{{{x^4}}}$ ,such that $f\left( 2 \right) = 0$ then, $f\left( x \right)$ is

SA

Q1 $\int\limits_0^1 {\cfrac{x}{{{x^2} + 1}}} dx$
SA Q2 $\int\limits_0^{\pi /2} {\sqrt {\sin \phi } {{\cos }^5}\phi d\phi }$ SA Q3 $\int\limits_0^1 {{{\sin }^{ - 1}}\left( {\cfrac{{2x}}{{1 + {x^2}}}} \right)dx}$
SA Q4 $\int\limits_0^2 {x\sqrt {x + 2} dx}$
SA Q5 $\int\limits_0^{\pi /2} {\cfrac{{\sin x}}{{1 + {{\cos }^2}x}}dx}$
SA Q6 $\int\limits_0^2 {\cfrac{{dx}}{{x + 4 - {x^2}}}}$
SA Q7 $\int\limits_{ - 1}^1 {\cfrac{{dx}}{{{x^2} + 2x + 5}}}$
SA Q8 $\int\limits_1^2 {\left( {\cfrac{1}{x} - \cfrac{1}{{2{x^2}}}} \right){e^{2x}}dx}$
SA Q9 The value of the integral $\int\limits_{1/3}^1 {\cfrac{{{{\left( {x - {x^3}} \right)}^{1/3}}}}{{{x^4}}}} dx$ is

SA Q10 If $f\left( x \right) = \int\limits_0^x {t\sin \,t\,dt}$ ,then $f'\left( x \right)$ is

SA

Q1 $\int\limits_0^{\pi /2} {{{\cos }^2}xdx}$
SA Q2 $\int\limits_0^\pi {\cfrac{{xdx}}{{1 + \sin x}}}$
SA Q2 $\int\limits_0^{2\pi } {{{\cos }^5}x} dx$
SA Q2 $\int\limits_0^{\pi /2} {\cfrac{{\sqrt {\sin x} }}{{\sqrt {\sin x + \sqrt {\cos x} } }}} dx$
SA Q3 $\int\limits_{}^{\pi /2} {\cfrac{{{{\sin }^{3/2}}xdx}}{{{{\sin }^{3/2}}x + {{\cos }^{3/2}}x}}}$
SA Q4 $\int\limits_0^{\pi /2} {\cfrac{{{{\cos }^5}xdx}}{{{{\sin }^5}x + {{\cos }^5}x}}}$
SA Q5 $\int\limits_{ - 5}^5 {\left| {x + 2} \right|} dx$
SA Q6 $\int\limits_2^8 {\left| {x - 5} \right|} dx$ SA Q7 $\int\limits_0^1 x {\left( {1 - x} \right)^n}dx$ SA Q8 $\int\limits_0^{\pi /4} {\log \left( {1 + \tan x} \right)dx}$
SA Q9 $\int\limits_0^2 {x\sqrt {2 - x} } dx$
SA Q10 $\int\limits_0^{\pi /2} {\left( {2\log \,sin\,x - log\,sin\,2x} \right)} dx$
SA Q11 $\int\limits_{ - \pi /2}^{\pi /2} {{{\sin }^2}x} dx$
SA Q11 $\int\limits_{ - \pi /2}^{\pi /2} {{{\sin }^7}x} dx$
SA Q15 $\int\limits_{0.}^{\pi /2} {\cfrac{{\sin x - \cos x}}{{1 + \sin x\cos x}}dx}$ SA Q17 $\int\limits_0^a {\cfrac{{\sqrt x }}{{\sqrt x + \sqrt {a - x} }}dx}$
SA Q18 $\int\limits_0^4 {\left| {x - 1} \right|dx}$
SA Q19 Show that$\int\limits_0^a {f\left( x \right)g\left( x \right)dx = 2\int\limits_0^a {f\left( x \right)dx} }$ ,

if f and g are defined as $f\left( x \right) = f\left( {a - x} \right)$ and $g\left( x \right) + g\left( {a - x} \right) = 4$
SA Q20 The value of $\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^3} + x\cos x + {{\tan }^5}x + 1} \right)dx}$ is

SA Q21 The value of $\int\limits_0^{\pi /2} {\log \left( {\cfrac{{4 + 3\sin x}}{{4 + 3\cos x}}} \right)dx}$ is

SA Q216 $\int\limits_0^\pi {\log \left( {1 + \cos x} \right)dx}$
SA

Q1 $\cfrac{{\cos 2x - \cos 2\alpha }}{{\cos x - \cos \alpha }}$
SA

Q1 $\cfrac{{2x}}{{1 + {x^2}}}$ SA Q2 $\cfrac{{{{\left( {\log x} \right)}^2}}}{x}$ SA Q3 $\cfrac{1}{{x + x\log x}}$ SA Q4 $\sin x\sin \left( {\cos x} \right)$ SA Q5 $\sin \left( {ax + b} \right)\cos \left( {ax + b} \right)$ SA Q6 $\sqrt {ax + b}$ SA Q7 $x\sqrt {x + 2}$ SA Q8 $x\sqrt {1 + 2{x^2}}$ SA Q9 $\left( {4x + 2} \right)\sqrt {{x^2} + x + 1}$ SA Q10 $\cfrac{1}{{x - \sqrt x }}$ SA Q11 $\cfrac{x}{{\sqrt {x + 4} }},x > 0$ SA Q12 ${\left( {{x^3} - 1} \right)^{1/3}}{x^5}$
SA Q13 $\cfrac{{{x^2}}}{{{{\left( {2 + 3{x^3}} \right)}^3}}}$ SA Q14 $\cfrac{1}{{x{{\left( {\log x} \right)}^m}}},x > 0$ SA Q15 $\cfrac{x}{{9 - 4{x^2}}}$ SA Q16 ${e^{2x + 3}}$ SA Q17 $\cfrac{x}{{{e^{{x^2}}}}}$ SA Q18 $\cfrac{{{e^{{{\tan }^{ - 1}}x}}}}{{1 + {x^2}}}$ SA Q19 $\cfrac{{{e^{2x}} - 1}}{{{e^{2x}} + 1}}$ SA Q20 $\cfrac{{{e^{2x}} - {e^{ - 2x}}}}{{{e^{2x}} + {e^{ - 2x}}}}$
SA Q21 ${\tan ^2}\left( {2x - 3} \right)$
SA Q22 ${\sec ^2}\left( {7 - 4x} \right)$
SA Q23 $\cfrac{{{{\sin }^{ - 1}}x}}{{\sqrt {1 - {x^2}} }}$ SA Q24 $\cfrac{{2\cos x - 3\sin x}}{{6\cos x + 4\sin x}}$ SA Q25 $\cfrac{1}{{{{\cos }^2}x{{\left( {1 - \tan x} \right)}^2}}}$
SA Q26 $\cfrac{{\cos \sqrt x }}{{\sqrt x }}$
SA Q27 $\sqrt {\sin 2x} \cos 2x$
SA Q28 $\cfrac{{\cos x}}{{\sqrt {1 + \sin x} }}$ SA Q29 $\cot x\log \sin x$
SA Q30 $\cfrac{{\sin x}}{{1 + \cos x}}$
SA Q31 $\cfrac{{\sin x}}{{{{\left( {1 + \cos x} \right)}^2}}}$
SA Q32 $\cfrac{1}{{1 + \cot x}}$
SA Q33 $\cfrac{1}{{1 - \tan x}}$
SA Q34 $\cfrac{{\sqrt {\tan x} }}{{\sin x\cos x}}$
SA Q35 $\cfrac{{{{\left( {1 + \log x} \right)}^2}}}{x}$
SA Q36 $\cfrac{{\left( {x + 1} \right){{\left( {x + \log x} \right)}^2}}}{x}$
SA Q37 $\cfrac{{{x^3}\sin \left( {{{\tan }^{ - 1}}{x^4}} \right)}}{{1 + {x^8}}}$
SA Q38 $\int {\left( {\cfrac{{10{x^9} + {{10}^x}{{\log }_e}10}}{{{x^{10}} + {{10}^x}}}} \right)} dx$equals

SA Q39 $\int {\cfrac{{dx}}{{{{\sin }^2}x{{\cos }^2}x}}}$equals

SA

Q1 ${\sin ^2}\left( {2x + 5} \right)$
SA Q2 $\sin 3x\cos 4x$
SA Q3 $\cos 2x\cos 4x\cos 6x$
SA Q4 ${\sin ^3}\left( {2x + 1} \right)$
SA Q5 ${\sin ^3}x{\cos ^3}x$
SA Q6 $\sin x\sin 2x\sin 3x$
SA Q7 $\sin 4x\sin 8x$
SA Q8 $\cfrac{{1 - \cos x}}{{1 + \cos x}}$
SA Q9 $\cfrac{{\cos x}}{{1 + \cos x}}$
SA Q10 ${\sin ^4}x$
SA Q11 ${\cos ^4}2x$
SA Q12 $\cfrac{{{{\sin }^2}x}}{{1 + \cos x}}$
SA Q14 $\cfrac{{\cos x - \sin x}}{{1 + \sin 2x}}$
SA Q15 ${\tan ^3}2x\sec 2x$
SA Q16 ${\tan ^4}x$
SA Q17 $\cfrac{{{{\sin }^3}x + {{\cos }^3}x}}{{{{\sin }^2}x{{\cos }^2}x}}$
SA Q18 $\cfrac{{\cos 2x + 2{{\sin }^2}x}}{{{{\cos }^2}x}}$
SA Q19 $\cfrac{1}{{\sin x{{\cos }^3}x}}$
SA Q20 $\cfrac{{\cos 2x}}{{{{\left( {\cos x + \sin x} \right)}^2}}}$
SA Q21 ${\sin ^{ - 1}}\left( {\cos x} \right)$
SA Q22 $\cfrac{1}{{\cos \left( {x - a} \right)\cos \left( {x - b} \right)}}$
SA Q23 $\int {\cfrac{{{{\sin }^2}x - {{\cos }^2}x}}{{si{n^2}x{{\cos }^2}x}}dx}$ is equals to

SA Q24 $\int {\cfrac{{{e^x}\left( {1 + x} \right)}}{{{{\cos }^2}\left( {{e^x}x} \right)}}dx}$ equals

SA

Q1 $\cfrac{{3{x^2}}}{{{x^6} + 1}}$ SA Q2 $\cfrac{1}{{\sqrt {1 + 4{x^2}} }}$
SA Q3 $\cfrac{1}{{\sqrt {{{\left( {2 - x} \right)}^2} + 1} }}$
SA Q4 $\cfrac{1}{{\sqrt {9 - 25{x^2}} }}$ SA Q5 $\cfrac{{3x}}{{1 + 2{x^4}}}$
SA Q6 $\cfrac{{{x^2}}}{{1 - {x^6}}}$ SA Q7 $\cfrac{{x - 1}}{{\sqrt {{x^2} - 1} }}$ SA Q8 $\cfrac{{{x^2}}}{{\sqrt {{x^6} + {a^2}} }}$
SA Q9 $\cfrac{{{{\sec }^2}x}}{{\sqrt {{{\tan }^2}x + 4} }}$
SA Q10 $\cfrac{1}{{\sqrt {{x^2} + 2x + 2} }}$
SA Q11 $\cfrac{1}{{9{x^2} + 6x + 5}}$ SA Q12 $\cfrac{1}{{\sqrt {7 - 6x - {x^2}} }}$
SA Q13 $\cfrac{1}{{\sqrt {\left( {x - 1} \right)\left( {x - 2} \right)} }}$ SA Q14 $\cfrac{1}{{\sqrt {8 + 3x - {x^2}} }}$
SA Q15 $\cfrac{1}{{\sqrt {\left( {x - a} \right)\left( {x - b} \right)} }}$
SA Q16 $\cfrac{{4x + 1}}{{\sqrt {2{x^2} + x - 3} }}$
SA Q17 $\cfrac{{x + 2}}{{\sqrt {{x^2} - 1} }}$
SA Q18 $\cfrac{{5x - 2}}{{1 + 2x + 3{x^2}}}$
SA Q19 . $\cfrac{{6x + 7}}{{\sqrt {\left( {x - 5} \right)\left( {x - 4} \right)} }}$
SA Q20 $\cfrac{{x + 2}}{{\sqrt {4x - {x^2}} }}$
SA Q21 $\cfrac{{x + 2}}{{\sqrt {{x^2} + 2x + 3} }}$
SA Q22 $\cfrac{{x + 3}}{{{x^2} - 2x - 5}}$
SA Q23 $\cfrac{{5x + 3}}{{\sqrt {{x^2} + 4x + 10} }}$
SA Q24 $\int {\cfrac{{dx}}{{{x^2} + 2x + 2}}}$ equals

SA Q25 $\int {\cfrac{{dx}}{{\sqrt {9x - 4{x^2}} }}}$ equals

SA

Q1 $\cfrac{x}{{\left( {x + 1} \right)\left( {x + 2} \right)}}$
SA Q2 $\cfrac{1}{{{x^2} - 9}}$
SA Q3 $\cfrac{{3x - 1}}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)}}$
SA Q4 $\cfrac{x}{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)}}$
SA Q5 $\cfrac{{2x}}{{{x^2} + 3x + 2}}$ SA Q6 $\cfrac{{1 - {x^2}}}{{x\left( {1 - 2x} \right)}}$
SA Q7 $\cfrac{x}{{\left( {{x^2} + 1} \right)\left( {x - 1} \right)}}$
SA Q8 $\cfrac{x}{{{{\left( {x - 1} \right)}^2}\left( {x + 2} \right)}}$
SA Q9 $\cfrac{{3x + 5}}{{{x^3} - {x^2} - x + 1}}$
SA Q10 $\cfrac{{2x - 3}}{{\left( {{x^2} - 1} \right)\left( {2x + 3} \right)}}$
SA Q11 $\cfrac{{5x}}{{\left( {x + 1} \right)\left( {{x^2} - 4} \right)}}$
SA Q12 $\cfrac{{{x^3} + x + 1}}{{{x^2} - 1}}$
SA Q13 $\cfrac{2}{{\left( {1 - x} \right)\left( {1 + {x^2}} \right)}}$
SA Q14 $\cfrac{{3x - 1}}{{{{\left( {x + 2} \right)}^2}}}$
SA Q15 $\cfrac{1}{{{x^4} - 1}}$
SA Q16 $\cfrac{1}{{x\left( {{x^n} + 1} \right)}}$
[Hint : multiply numerator and denominator by ${x^{n - 1}}$ and Put ${x^n} = t$ ]
SA Q17 $\cfrac{{\cos x}}{{\left( {1 - \sin x} \right)\left( {2 - \sin x} \right)}}$ [Hint : Put $\sin x = t$]
SA Q18 $\cfrac{{\left( {{x^2} + 1} \right)\left( {{x^2} + 2} \right)}}{{\left( {{x^2} + 3} \right)\left( {{x^2} + 4} \right)}}$
SA Q19 $\cfrac{{2x}}{{\left( {{x^2} + 1} \right)\left( {{x^2} + 3} \right)}}$
SA Q20 $\cfrac{1}{{x\left( {{x^4} - 1} \right)}}$
SA Q21 $\cfrac{1}{{\left( {{e^x} - 1} \right)}}$ [Hint : Put ${e^x} = t$ ]
SA Q23 $\int {\cfrac{{x\,dx}}{{\left( {x - 1} \right)\left( {x - 2} \right)}}}$ equals

SA Q23 $\int {\cfrac{{dx}}{{x\left( {{x^2} + 1} \right)}}}$ equals

SA

Q1 $x\sin x$ SA Q2 $x\sin 3x$
SA Q3 ${x^2}{e^x}$ SA Q4 $x\log x$ SA Q5 $x\log 2x$ SA Q6 ${x^2}\log x$ SA Q7 $x{\sin ^{ - 1}}x$
SA Q8 $x{\tan ^{ - 1}}x$
SA Q9 $x{\cos ^{ - 1}}x$
SA Q10 ${\left( {{{\sin }^{ - 1}}x} \right)^2}$
SA Q11 $\cfrac{{x{{\cos }^{ - 1}}x}}{{\sqrt {1 - {x^2}} }}$
SA Q12 $x{\sec ^2}x$
SA Q13 ${\tan ^{ - 1}}x$
SA Q14 $x{\left( {\log x} \right)^2}$
SA Q15 $\left( {{x^2} + 1} \right)\log x$ SA Q16 ${e^x}\left( {\sin x + \cos x} \right)$ SA Q17 $\cfrac{{x{e^x}}}{{{{\left( {1 + x} \right)}^2}}}$
SA Q18 ${e^x}\left( {\cfrac{{1 + \sin x}}{{1 + \cos x}}} \right)$
SA Q19 ${e^x}\left( {\cfrac{1}{x} - \cfrac{1}{{{x^2}}}} \right)$
SA Q20 . $\cfrac{{\left( {x - 3} \right){e^x}}}{{{{\left( {x - 1} \right)}^3}}}$
SA Q21 . ${e^{2x}}\sin x$
SA Q22 ${\sin ^{ - 1}}\left( {\cfrac{{2x}}{{1 + {x^2}}}} \right)$
SA Q23 $\int {{x^2}{e^{{x^3}}}dx}$ equals

SA Q24 $\int {{e^x}\sec x\left( {1 + \tan x} \right)dx}$ equals

SA

Q1 $\sqrt {4 - {x^2}}$
SA Q2 $\sqrt {1 - 4{x^2}}$
SA Q3 $\sqrt {{x^2} + 4x + 6}$
SA Q4 $\sqrt {{x^2} + 4x + 1}$
SA Q5 $\sqrt {1 - 4x - {x^2}}$
SA Q6 $\sqrt {{x^2} + 4x - 5}$
SA Q8 $\sqrt {1 + 3x - {x^2}}$
SA Q8 $\sqrt {{x^2} + 3x}$
SA Q9 $\sqrt {1 + \cfrac{{{x^2}}}{9}}$
SA Q10 $\int {\sqrt {1 + {x^2}} dx}$ is equal to

SA Q11 $\int {\sqrt {{x^2} - 8x + 7} dx}$ is equal to

SA

Q1 $\int\limits_a^b {x\,dx}$
SA Q1 $\int\limits_{ - 1}^1 {\left( {x + 1} \right)dx}$
SA Q2 $\int\limits_0^5 {\left( {x + 1} \right)\,dx}$
SA Q3 $\int\limits_2^3 {{x^2}\,dx}$
SA Q4 $\int\limits_1^4 {\left( {{x^2} - x} \right)dx}$
SA Q5 $\int\limits_{ - 1}^1 {{e^x}dx}$
SA Q6 $I = \int\limits_0^4 {\left( {x + {e^{2x}}} \right)dx}$
SA

Q2 $\int\limits_2^3 {\cfrac{1}{x}dx}$ SA Q3 $\int\limits_1^2 {\left( {4{x^3} - 5{x^2} + 6x + 9} \right)dx}$
SA Q4 $\int\limits_0^{\pi /4} {\sin 2x} \,dx$ SA Q5 $\int\limits_0^{\pi /2} {\cos 2x} \,dx$
SA Q6 $\int\limits_4^5 {{e^x}} dx$
SA Q7 $\int\limits_0^{\pi /4} {\tan x} dx$ SA Q8 $\int\limits_{\pi /6}^{\pi /4} {\cos ec\,x\,} dx$ SA Q9 $\int\limits_0^1 {\cfrac{{dx}}{{\sqrt {1 - {x^2}} }}}$
SA Q10 $\int\limits_0^1 {\cfrac{{dx}}{{1 + {x^2}}}}$ SA Q11 $\int\limits_2^3 {\cfrac{{dx}}{{{x^2} - 1}}}$
SA Q12 $\int\limits_0^{\pi /2} {\cos }^2x dx$
SA Q13 . $\int\limits_2^3 {\cfrac{{x\,dx}}{{{x^2} + 1}}}$
SA Q14 $\int\limits_0^1 {\cfrac{{2x + 3}}{{5{x^2} + 1}}dx}$
SA Q15 $\int\limits_0^1 x{e^{{x^2}}}dx$

SA Q16 $\int\limits_1^2 {\cfrac{{5{x^2}}}{{{x^2} + 4x + 3}}}$
SA Q17 $\int\limits_0^{\pi /4} {\left( {2{{\sec }^2}x + {x^3} + 2} \right)dx}$
SA Q18 $\int\limits_0^\pi {\left( {{{\sin }^2}\cfrac{x}{2} - {{\cos }^2}\cfrac{x}{2}} \right)dx}$
SA Q19 $\int\limits_0^2 {\cfrac{{6x + 3}}{{{x^2} + 4}}dx}$
SA Q20 $\int\limits_0^1 {\left( {x{e^x} + \sin \cfrac{{\pi x}}{4}} \right)dx}$
SA Q21 $\int\limits_1^{\sqrt 3 } {\cfrac{{dx}}{{1 + {x^2}}}}$ equals

SA Q22 $\int\limits_0^{2/3} {\cfrac{{dx}}{{4 + 9{x^2}}}}$ equals

SA

Q9 $\int {\left( {2{x^2} + {e^x}} \right)dx}$ SA

Q1 $\cfrac{1}{{x - {x^3}}}$
SA Q2 $\cfrac{1}{{\sqrt {x + a} + \sqrt {x + b} }}$
SA Q3 $\cfrac{1}{{x\sqrt {ax - {x^2}} }}$ [Hint : Put $x = \cfrac{a}{t}$ ]
SA Q4 $\cfrac{1}{{{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}$
SA Q5 $\cfrac{1}{{{x^{1/2}} + {x^{1/3}}}}$
[Hint : $\cfrac{1}{{{x^{1/2}} + {x^{1/3}}}} = \cfrac{1}{{{x^{1/3}}\left( {1 + {x^{1/6}}} \right)}},{\rm{Put}}\,x = {t^6}$ ]
SA Q7 $\cfrac{{5x}}{{\left( {x + 1} \right)\left( {{x^2} + 9} \right)}}$
SA Q7 $\cfrac{{\sin x}}{{\sin \left( {x - a} \right)}}$
SA Q8 $\cfrac{{{e^{5\log x}} - {e^{4\log x}}}}{{{e^{3\log x}} - {e^{2\log x}}}}$
SA Q9 $\cfrac{{\cos x}}{{\sqrt {4 - {{\sin }^2}x} }}$
SA Q10 $\cfrac{{{{\sin }^8}x - {{\cos }^8}x}}{{1 - 2{{\sin }^2}x{{\cos }^2}x}}$
SA Q11 $\cfrac{1}{{\cos \left( {x + a} \right)\cos \left( {x + b} \right)}}$
SA Q12 . $\cfrac{{{x^3}}}{{\sqrt {1 - {x^8}} }}$ SA Q13 $\cfrac{{{e^x}}}{{\left( {1 + {e^x}} \right)\left( {2 + {e^x}} \right)}}$
SA Q14 . $\cfrac{1}{{\left( {{x^2} + 1} \right)\left( {{x^2} + 4} \right)}}$
SA Q15 ${\cos ^3}x{e^{\log \sin x}}$
SA Q16 ${e^{3\log x}}{\left( {{x^4} + 1} \right)^{ - 1}}$
SA Q17 $f'\left( {ax + b} \right){\left[ {f\left( {ax + b} \right)} \right]^n}$
SA Q18 $\cfrac{1}{{\sqrt {{{\sin }^3}x\sin \left( {x + \alpha } \right)} }}$ SA Q19 $\cfrac{{{{\sin }^{ - 1}}\sqrt x - {{\cos }^{ - 1}}\sqrt x }}{{{{\sin }^{ - 1}}\sqrt x + {{\cos }^{ - 1}}\sqrt x }},x \in \left[ {0,1} \right]$
SA Q20 $\sqrt {\cfrac{{1 - \sqrt x }}{{1 + \sqrt x }}}$ SA Q21 $\cfrac{{2 + \sin 2x}}{{1 + \cos 2x}}{e^x}$ SA Q22 . $\cfrac{{{x^2} + x + 1}}{{{{\left( {x + 1} \right)}^2}\left( {x + 2} \right)}}$
SA Q23 ${\tan ^{ - 1}}\sqrt {\cfrac{{1 - x}}{{1 + x}}}$
SA Q24 . $\cfrac{{\sqrt {{x^2} + 1} \left[ {\log \left( {{x^2} + 1} \right) - 2\log x} \right]}}{{{x^4}}}$
SA Q25 $\int\limits_{\pi /2}^\pi {{e^x}\left( {\cfrac{{1 - \sin x}}{{1 - \cos x}}} \right)} dx$
SA Q26 $\int\limits_0^{\pi /4} {\cfrac{{\sin x\cos x}}{{{{\cos }^4}x + {{\sin }^4}x}}dx}$

SA Q27 $\int\limits_0^{\pi /2} {\cfrac{{{{\cos }^2}xdx}}{{{{\cos }^2}x + 4{{\sin }^2}x}}}$
SA Q28 $\int\limits_{\pi /6}^{\pi /3} {\cfrac{{\sin x + \cos x}}{{\sqrt {\sin 2x} }}dx}$
SA Q29 $\int\limits_0^1 {\cfrac{{dx}}{{\sqrt {1 + x} - \sqrt x }}}$
SA Q30 $\int\limits_0^{\pi /4} {\cfrac{{{\mathop{\rm sinx}\nolimits} + cosx}}{{9 + 16\sin 2x}}} dx$
SA Q31 $\int\limits_0^{\pi /2} {\sin 2x{{\tan }^{ - 1}}\left( {\sin x} \right)dx}$
SA Q32 $\int\limits_0^\pi {\cfrac{{x\tan x}}{{\sec x + {\mathop{\rm tanx}\nolimits} }}dx}$
SA Q33 $\int\limits_1^4 {\left[ {\left| {x - 1} \right| + \left| {x - 2} \right| + \left| {x - 3} \right|} \right]} dx$
SA Q34 . $\int\limits_1^3 {\cfrac{{dx}}{{{x^2}\left( {x + 1} \right)}} = \cfrac{2}{3} + \log \cfrac{2}{3}}$
SA Q35 $\int\limits_0^1 {x{e^x}dx} = 1$
SA Q36 $\int\limits_{ - 1}^1 {{x^{17}}{{\cos }^4}x} dx = 0$
SA Q37 $\int\limits_0^{\pi /2} {{{\sin }^3}xdx = \cfrac{2}{3}}$
SA Q38 $\int\limits_0^{\pi /4} {2{{\tan }^3}xdx = 1 - \log 2}$
SA Q39 $\int\limits_0^1 {{{\sin }^{ - 1}}xdx = \cfrac{\pi }{2} - 1}$
SA Q40 Evaluate $\int\limits_0^1 {{e^{2 - 3x}}} dx$ as a limit of a sum.
SA Q41 $\int {\cfrac{{dx}}{{{e^x} + {e^{ - x}}}}}$is equal to

SA Q42 $\int {\cfrac{{\cos 2x}}{{{{\left( {\sin x + \cos x} \right)}^2}}}} dx$ is equal to

SA Q43 If $f\left( {a + b - x} \right) = f\left( x \right)$ , then $\int\limits_a^b {xf\left( x \right)dx}$ is equal to

SA Q44 The value of $\int\limits_0^1 {{{\tan }^{ - 1}}\left( {\cfrac{{2x - 1}}{{1 + x - {x^2}}}} \right)} dx$ is

SA