Question

help me pls: ∫cos ³(x)sin(x)dx

Answer 1

Hello! :)

$y = -\frac{cos^{4}(x)}{4} + C$

Use u-substitution to solve for the indefinite integral:

$\int cos^{3}(x)sin(x)dx$

Allow "u" to be the expression with an exponent:

$u = cos(x)\\\\du = -sin(x)dx$

$-du = sin(x)dx$

In the integral, we are missing a negative symbol (du = -sin(x)), so we can adjust the integral to accommodate this.

Substitute "u" for cos(x) and du for -sin(x):

$-\int u^{3}du$

Use the integral power rule to solve:

$\int x^{n} = \frac{x^{n + 1}}{n + 1}$

$-\int u^{3}du = -[\frac{u^{4}}{4} ]$

Add the constant "C" as this is an indefinite integral:

$= -[\frac{u^{4}}{4} ] + C$

Substitute in the value of u (cos(x)) into the equation:

$= -\frac{cos^{4}(x)}{4} + C$

And you're done!