e^{x/2} 2x (1+4 2x) dx is equal to

Correct Answer is : None of these

Correct Answer is : None of these

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CGPET 2019 Solved Paper

Section: Mathematics

Question : 111 of 150

Marks: +1, -0

\int e^{\frac{x}{2}} \sec 2x (1+4\tan 2x) dx

is equal to

Solution:

I=\int e^{\frac{x}{2}} \sec 2x (1+4\tan 2x) dx

=\int e^{\frac{x}{2}} (\sec 2x+4\tan 2x \cdot \sec 2x) dx

Let

\frac{x}{2}=t,\ x=2t,\ dx=2dt

\therefore I=2\int e^{t} (\sec 4t+4\tan 4t \cdot \sec 4t) dt

.......(i)

We know,

\int e^{x} (f(x)+f'(x)) dx

=e^{x}f(x)+C

.......(ii)

and

\frac{d}{dx}(\sec 4x)=4\sec 4x \tan 4x

From Eqs. (i) and (ii), we get

=2[e^{t}\sec 4t+C']=2e^{t}\sec 4t+2C'

Put the value of

t

, we get

=2e^{x/2}\sec 4 \cdot \left(\frac{x}{2}\right)+2c'

=2e^{x/2}\sec 2x+C

[

Here

2C'=C]

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