Mary Hammonds

2021-12-27

Evaluate the following integrals:

$\int {\mathrm{cos}}^{2}x\mathrm{sin}xdx$

Pansdorfp6

Beginner2021-12-28Added 27 answers

Step 1

In some cases, an integral can be simplified to a standard integral with an appropriate substitution. For example, the integral$\int {\mathrm{sin}}^{2}x\mathrm{cos}xdx$ can be converted to $\int {u}^{2}du$ with the substitution $u=\mathrm{sin}x$ .

For the given problem, use the integral$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1},n\ne -1$ . Find an appropriate substitution for the given integral to simplify the integrand to a simpler integrand.

Step 2

Integral to be computed is$\int {\mathrm{cos}}^{2}x\mathrm{sin}xdx$ . Use the substitution, $\mathrm{cos}x=u$ . Differentiating this, gives $-\mathrm{sin}xdx=du$ . Apply this substitution and integrate using information from step 1.

$\int {\mathrm{cos}}^{2}x\mathrm{sin}xdx=\int {u}^{2}(-du)$

$=-\int {u}^{2}du$

$=-\frac{{u}^{3}}{3}+C$

$=-\frac{{\mathrm{cos}}^{3}x}{3}+C$

Hence, the integral is equal to$-\frac{{\mathrm{cos}}^{3}x}{3}+C$ .

In some cases, an integral can be simplified to a standard integral with an appropriate substitution. For example, the integral

For the given problem, use the integral

Step 2

Integral to be computed is

Hence, the integral is equal to

braodagxj

Beginner2021-12-29Added 38 answers

karton

Expert2022-01-04Added 613 answers

Add C

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