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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.2
Integrate the functions in Exercises 23 to 59.
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Question : 27 of 271
Marks: +1, -0
sin⁡(ax+b)cos⁡(ax+b)\sin(ax + b) \cos(ax + b)
Solution:  
Let I=∫sin⁡(ax+b)cos⁡(ax+b)dxI = \int \sin(ax + b) \cos(ax + b) dx
Put sin⁡(ax+b)=t\sin(ax + b) = t ⇒acos⁡(ax+b)dx=dt\Rightarrow a \cos(ax + b) dx = dt
∴I=1a∫tdt=1a⋅t22+C\therefore I = \frac{1}{a} \int t dt = \frac{1}{a} \cdot \frac{t^2}{2} + C =12at2+C= \frac{1}{2a} t^2 + C =12asin⁡2(ax+b)+C= \frac{1}{2a} \sin^2(ax+b) + C
or, put cos⁡(ax+b)=t\cos(ax + b) = t ⇒−asin⁡(ax+b)dx=dt\Rightarrow -a \sin(ax + b) dx = dt
∴I=−1a∫tdt=−1a⋅t22+C\therefore I = -\frac{1}{a} \int t dt = -\frac{1}{a} \cdot \frac{t^2}{2} + C =−cos⁡2(ax+b)2a+C= \frac{-\cos^2(ax+b)}{2a} + C
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