Evaluate the integral.

{eq}\int_{0}^{\frac{\pi}{2}} \sin^7(\theta) \cos^5(\theta) \, \mathrm{d} \theta {/eq}

Question:

Definite Integral by Substitution:

If the given integral is of the form:

{eq}\int_{a}^{b}f'(g(x))g'(x)dx {/eq}

then we use the substitution:

{eq}g(x)=t\\ g'(x)dx=dt {/eq}

Now if the function:

{eq}g(x) {/eq}

Is invertible in the given interval of limits of integration, the new limits will be:

{eq}g(a),g(b) {/eq}

So the integral will be:

{eq}\int_{g(a)}^{g(b)}f'(t)dt {/eq}

Answer and Explanation: 1