Evaluate the following integrals: ∫cos2xsin⁡xdx

Question

Evaluate the following integrals:  ∫cos2xsin⁡xdx

Pansdorfp6 · Accepted Answer

Step 1
In some cases, an integral can be simplified to a standard integral with an appropriate substitution. For example, the integral

\int \sin^{2} x \cos x d x

can be converted to

\int u^{2} d u

with the substitution

u = \sin x

.
For the given problem, use the integral

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}, n \neq - 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 \cos^{2} x \sin x d x

. Use the substitution,

\cos x = u

. Differentiating this, gives

- \sin x d x = d u

. Apply this substitution and integrate using information from step 1.

\int \cos^{2} x \sin x d x = \int u^{2} (- d u)

= - \int u^{2} d u

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

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

Hence, the integral is equal to

- \frac{\cos^{3} x}{3} + C

.

braodagxj · Accepted Answer

∫cos2(x)sin⁡(x)dx  =−∫u2du  ∫u2du  =u33  −∫u2du  =−u33  =−cos3(x)3  ∫cos2(x)sin⁡(x)dx  =−cos3(x)3+C

karton · Accepted Answer

∫cos⁡(x)2sin⁡(x)dx ∫−t2dt −∫t2dt −t33 −cos⁡(x)33 Add C Answer: −cos⁡(x)33+C

Evaluate the following integrals: \int \cos^{2}x\sin x dx