e^x≤ft(1+ x/1+ x) - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.6
Integrate the functions in Exercises 134 to 155.

Question : 151 of 271

Marks: +1, -0

e^x\left(\frac{1+\sin x}{1+\cos x}\right)

Solution:

Let

I=\int\limits e^x\left(\frac{1+\sin x}{1+\cos x}\right)dx

I=\int\limits e^x\left[\frac{1+2\sin\frac{x}{2}\cos\frac{x}{2}}{2\cos^2\frac{x}{2}}\right]dx

=\int\limits e^x\left[\frac{1}{2}\sec^2\frac{x}{2}+\tan\frac{x}{2}\right]dx

=\int\limits e^x\left[\tan\frac{x}{2}+\frac{1}{2}\sec^2\frac{x}{2}\right]dx

=\int\limits e^x\left[\tan\frac{x}{2}+\frac{d}{dx}\left(\tan\frac{x}{2}\right)\right]dx

=e^x\cdot\tan\frac{x}{2}+C

\left[\because\int\limits e^x\left(f(x)+f'(x)\right)dx=e^x f(x)+C\right]

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