Integrals — Class 12 Maths Solution

ncert exercise SA NCERT,ex.7.2,Q.5,Page 304
Question

$\sin \left( {ax + b} \right)\cos \left( {ax + b} \right)$

Step-by-step Solution

.: Let$I = \int {\sin \left( {ax + b} \right)\cos \left( {ax + b} \right)dx}$

Put $\sin \left( {ax + b} \right) = t$ $\Rightarrow$ $a\cos \left( {ax + b} \right)dx = dt$

$\therefore$ $I = \cfrac{1}{a}\int t dt = \cfrac{1}{a} \cdot \cfrac{{{t^2}}}{2} + C = \cfrac{1}{{2a}}{t^2} + C = \cfrac{1}{{2a}}{\sin ^2}\left( {ax + b} \right) + C$

Aliter : Put $\cos \left( {ax + b} \right) = t$ $\Rightarrow$ $- a\sin \left( {ax + b} \right)dx = dt$

$\therefore$ $I = \cfrac{{ - 1}}{a}\int t dt = \cfrac{{ - 1}}{a} \cdot \cfrac{{{t^2}}}{2} + C = \cfrac{{ - {{\cos }^2}\left( {ax + b} \right)}}{{2a}} + C$

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NCERT & Exemplar solution for CBSE Class 12 Mathematics, Integrals. Curated by Sachin Sharma. Free for all students.