Math Help - Integration by substitution

1. Integration by substitution

I'm having difficulty with solving this:

$\int_{0}^{\frac{\pi}{2}} \frac{cos\:t}{\sqrt{3+sin\:t}}\: dt$

I'd greatly appreciate any help!

2. Re: Integration by substitution

Originally Posted by maybealways
I'm having difficulty with solving this:

$\int_{0}^{\frac{\pi}{2}} \frac{cos\:t}{\sqrt{3+sin\:t}}\: dt$

I'd greatly appreciate any help!
Make the substitution $\displaystyle u = 3 + \sin{t} \implies du = \cos{t}\,dt$, and also $\displaystyle u(0) = 3$ and $\displaystyle u\left(\frac{\pi}{2}\right) = 4$ and the integral becomes

$\displaystyle \int_3^4{\frac{1}{\sqrt{u}}\,du} = \int_3^4{u^{-\frac{1}{2}}\,du}$.

Can you go from here?

3. Re: Integration by substitution

I'm still having difficulty with solving the rest of the equation ^^;;;

Also, why does $\displaystyle u\left(\frac{\pi}{2}\right)$?

4. Re: Integration by substitution

Like Prove it said the substitution $u=3+\sin(t) \Rightarrow du=\cos(t)dt$ is very useful here.
Because you have a definite integral you can change the integration limits by using the given substitution.
The original integration limits are in function of $t$ so the new one has to be in function of $u$ so:
$t=\frac{\pi}{2} \Rightarrow u=3+\sin\left(\frac{\pi}{2}\right)=3+1=4$
$t=0 \Rightarrow u=3+\sin(0)=3$
So the new integration limits (in function of $u$) are 3 and 4.

To solve the integral use the rule:
$\int_{a}^{b} x^{n} = \left[\frac{x^{n+1}}{n+1}\right]_{a}^{b}$ and $n\neq -1$

Note:
Changing the integration limits isn't necessary, you can also hold the original integration limits, but afterwards you have to do the back-substitution then.

5. Re: Integration by substitution

Originally Posted by maybealways
I'm still having difficulty with solving the rest of the equation ^^;;;

Also, why does $\displaystyle u\left(\frac{\pi}{2}\right)$?
You are making a substitution $\displaystyle u(t) = 3 + \sin{t}$, so the notation $\displaystyle u(0)$ means $\displaystyle u$ evaluated at $\displaystyle t = 0$, and similarly $\displaystyle u\left(\frac{\pi}{2}\right)$ means $\displaystyle u$ evaluated at $\displaystyle t = \frac{\pi}{2}$. As Siron said, by changing the terminals to $\displaystyle u$ values, you don't need to convert your integral back to a function of $\displaystyle t$ before substituting your terminals.

6. Re: Integration by substitution

^Oh! I see That makes sense (in regards to the the terminal part).

But, I'm still stuck after everything has been substituted :/ Do I equate both sides separately?

7. Re: Integration by substitution

Originally Posted by maybealways
^Oh! I see That makes sense (in regards to the the terminal part).

But, I'm still stuck after everything has been substituted :/ Do I equate both sides separately?
Equate both sides of what?

8. Re: Integration by substitution

This: $\int_3^4{\frac{1}{\sqrt{u}}\,du} = \int_3^4{u^{-\frac{1}{2}}\,du}$

9. Re: Integration by substitution

Originally Posted by maybealways
This: $\int_3^4{\frac{1}{\sqrt{u}}\,du} = \int_3^4{u^{-\frac{1}{2}}\,du}$
They're two steps of the integration, not an "equation" to solve. Now you need to actually perform the integration using the rule given by Siron in post #4.