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Compute the integral using trig substitution

imattxc

New member
Joined
Oct 25, 2014
Messages
2
I need to correct the following problem using trig substitution.
eqn4315.png


I keep getting stuck at:

eqn4315.png

If I dont use trig sub and just use u-sub I get like 2/15, but I think I need to answer using trig sub. Thanks for any help you can offer!
 

Subhotosh Khan

Super Moderator
Staff member
Joined
Jun 18, 2007
Messages
18,086
I need to correct the following problem using trig substitution.
View attachment 4649


I keep getting stuck at:

View attachment 4648

If I dont use trig sub and just use u-sub I get like 2/15, but I think I need to answer using trig sub. Thanks for any help you can offer!


sin[SUP]3[/SUP](Θ) * cos[SUP]2[/SUP](Θ) = sin(Θ) * [1-cos[SUP]2[/SUP](Θ)] * cos[SUP]2[/SUP](Θ) = sin(Θ) * cos[SUP]2[/SUP](Θ) - sin(Θ) * cos[SUP]4[/SUP](Θ) .... Now integrate....
 

HallsofIvy

Elite Member
Joined
Jan 27, 2012
Messages
4,716
Typically, if you have an odd power of sine or cosine you can factor one out, to use with the "dx", then use \(\displaystyle sin^2(x)+ cos^2(x) = 1\), so that either \(\displaystyle sin^2(x)= 1- cos^2(x)\) or \(\displaystyle cos^2(x)= 1- sin^2(x)\) to reduce the remaining even power. Here, you have \(\displaystyle \int_0^{\pi/2} sin^3(\theta) cos^2(\theta)d\theta\) which we can write as \(\displaystyle \int_0^{\pi/2} sin^2(\theta) cos^2(\theta) (sin(\theta) d\theta)= \int_0^{\pi/2} (1- cos^2(\theta))cos^2(\theta)(sin(\theta)d\theta)= \int_0^{\pi/2} (cos^2(\theta)- cos^4(\theta))(sin(\theta)d\theta)\).

Now make the substitution \(\displaystyle u= cos(\theta)\).
 
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