# Thread: integration questions stuck :(

1. ## integration questions stuck :(

Hey guy i gt this questions on integration which i am not sure how to approach. I have uploaded the question via jpg file.

For the first questions i attempted to intergrate by parts and let u=e^tanx
du= tane^tanx.dx, dv/dx = cos^2x and v = (sin^3x)/3, am i correct so far?

As a results i was left with:

e^tanx . (sin^3x/3) - Int (sin^3X/3) . tane^tanx

what do i do now if in fact i am on the correct path i would assume to integrate it again but that seems too difficult and deafeats the purpose of the IBP. Was i supposed to use IBP? Any help would be greatly appreciated.

2. Originally Posted by Latkan
Hey guy i gt this questions on integration which i am not sure how to approach. I have uploaded the question via jpg file.

For the first questions i attempted to intergrate by parts and let u=e^tanx
du= tane^tanx.dx, dv/dx = cos^2x and v = (sin^3x)/3, am i correct so far?

As a results i was left with:

e^tanx . (sin^3x/3) - Int (sin^3X/3) . tane^tanx

what do i do now if in fact i am on the correct path i would assume to integrate it again but that seems too difficult and deafeats the purpose of the IBP. Was i supposed to use IBP? Any help would be greatly appreciated.
This looks like a simple substituion

$\displaystyle \int^{\pi/4}_{0} e^{\tan x} \sec^2(x)$

Because $\displaystyle \sec^2 x = \frac{1}{\cos^2 x}$

Let $\displaystyle u = \tan x$. Hence $\displaystyle du = \sec^2 x\,dx$

Now to work out our limits in terms of u. For this simply sub the limit into our expression for u.

$\displaystyle \text { Lower Limit } = \tan 0 = 0$

$\displaystyle \text { Higher Limit } = \tan \left(\frac{\pi}{4}\right) = 1$

Now we can put all these expressions into the integral to give:

$\displaystyle \int^1_0 \, e^u\,du$

Which is a very simple integral and can be solved directly because we changed the limits