Mock Test 2 — Indefinite Integration

Evaluate the following algebraic rational fraction integral: $\int \frac{x^2 - 1}{x^4 + x^2 + 1} \, dx$.

Solve the rational trigonometric fraction integral $\int \frac{1}{1 + 3\sin^2 x} \, dx$:

Evaluate the following algebraic integration form: $\int \frac{x^2}{(x\sin x + \cos x)^2} \, dx$.

The integral value for the Euler-algebraic substitution form $\int \frac{1}{x \sqrt{1 - x^5}} \, dx$ matches:

Solve the rational fraction containing a linear radical term: $\int \frac{1}{(x-1)\sqrt{x+2}} \, dx$.

Evaluate the specific algebraic substitution integral form $\int \frac{x^7}{(1-x^4)^2} \, dx$:

Find the value of the structural trigonometric product integral: $\int \frac{\cos 4x - 1}{\cot x - \tan x} \, dx$.

Evaluate the symmetric algebraic transformation form $\int \frac{1}{x^4 + 1} \, dx$:

Solve the following inverse trigonometric substitution integral: $\int \frac{\sin^{-1}\sqrt{x} - \cos^{-1}\sqrt{x}}{\sin^{-1}\sqrt{x} + \cos^{-1}\sqrt{x}} \, dx$.

If $\int \frac{1}{x^3 + x^5} \, dx = \frac{A}{x^2} + B\ln|x| + C\ln|1+x^2| + V$, then the values of the constant coefficients are:

If the substitution integral satisfies $\int \frac{x^3}{\sqrt{1+x^2}} \, dx = \frac{1}{k}(1+x^2)^{3/2} - \sqrt{1+x^2} + C$, find the value of the integer denominator $k$.

Find the total number of distinct linear factors that appear in the denominator of the partial fraction decomposition for $R(x) = \frac{1}{x^4 - 5x^2 + 4}$.

If $\int e^x \left[ \frac{x-1}{x^2} \right] \, dx = \frac{e^x}{k \cdot x} + C$, find the value of the integer scalar parameter $k$.

Assertion (A): The algebraic substitution integral $\int \frac{1}{x^2 - a^2} \, dx$ can be evaluated by using the algebraic substitution $x = a\sin\theta$.
Reason (R): Trigonometric substitutions use standard identities to simplify quadratic radical fields like $\sqrt{a^2 - x^2}$ into single terms.

Assertion (A): To evaluate rational functions of the type $\int \frac{1}{a + b\cos x} \, dx$, using the Weierstrass half-angle transformation $t = \tan(x/2)$ is an effective method.
Reason (R): This substitution transforms trigonometric rational functions into standard algebraic fractions that can be solved using standard quadratic forms.