Question

Find each integral. $$\int \sin ^{10} x \cos x d x$$

   Find each integral.
$$\int \sin ^{10} x \cos x d x$$

Calculus with Applications

Calculus with Applications

Margaret L. Lial •… 11th Edition

Chapter 13, Problem 9

Instant Answer

Solved by Expert Alex Johnson

02/06/2020

Step 1

We let $u = \sin x$. This gives us $du = \cos x \, dx$. Show more…

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Find each integral. $$\int \sin ^{10} x \cos x d x$$

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Key Concepts

u-Substitution

This integration technique involves changing the variable of integration to simplify the integral. By letting u equal a function inside the integrand (in this case, u = sin x), the differential du becomes easier to work with (du = cos x dx), effectively reducing the integral to a standard form.

Power Rule for Integration

After substitution, the integral often becomes a simple power function, such as ? u^n du. The power rule for integration is then applied, stating that the integral of u^n is u^(n+1)/(n+1) for n ? -1, which is fundamental in finding antiderivatives of polynomial expressions.

Trigonometric Integrals

The integration of functions involving powers of trigonometric functions, like sin^n x and cos x, often requires specialized techniques such as u-substitution or using trigonometric identities. Recognizing the structure of the integrand and applying appropriate methods is key to simplifying and evaluating the integral.

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