Evaluate the integral.

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


Evaluate the integral.

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

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

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The given integral is:

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

Use the substitution:


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Learn more about this topic:

How to Solve Integrals Using Substitution


Chapter 13 / Lesson 5

Explore the steps in integration by substitution. Learn the importance of integration with the chain rule and see the u-substitution formula with various examples.

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