1. ## Trig substitution

How can we find the integral $\displaystyle \int_{0}^{\pi}\cos{x}\sin{x}\,{dx}$ with the substitution $\displaystyle u = sinx$?

I know how to find the integral, but the limits become the same sin(0) = sin(pi) = 0.

What should I do in general when a given substitution makes upper bound = lower bound?

2. Originally Posted by Hardwork
How can we find the integral $\displaystyle \int_{0}^{\pi}\cos{x}\sin{x}\,{dx}$ with the substitution $\displaystyle u = sinx$?

I know how to find the integral, but the limits become the same sin(0) = sin(pi) = 0.

What should I do in general when a given substitution makes upper bound = lower bound?
Using a different property, you can analyse the integrand as a function with period "pi", the (signed) area of which equals zero after each period.

3. Originally Posted by Hardwork
How can we find the integral $\displaystyle \int_{0}^{\pi}\cos{x}\sin{x}\,{dx}$ with the substitution $\displaystyle u = sinx$?

I know how to find the integral, but the limits become the same sin(0) = sin(pi) = 0.

What should I do in general when a given substitution makes upper bound = lower bound?
Is...

... so that, no matter which 'substitution' You do, the integral is 0...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

4. Okay. I do see how the integral is zero. I just want to know what to do in general when a substitution makes the limits the same.