Trig substitution

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April 30th 2011, 09:05 PM
Hardwork

Trig substitution

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

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?
April 30th 2011, 09:21 PM
TheChaz

Quote:

Originally Posted by Hardwork

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

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.
April 30th 2011, 09:24 PM
chisigma

Quote:

Originally Posted by Hardwork

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

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...

http://quicklatex.com/cache3/ql_0a43...4bcb647_l3.png

http://quicklatex.com/cache3/ql_5801...1eb2dab_l3.png

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

Kind regards

$\chi$ $\sigma$
May 2nd 2011, 01:32 AM
Hardwork

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.