Question

Help me pls: ∫cos ³(x)sin(x)dx

Answer 1

Hello! :)

`y = -(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) = (x^(n + 1))/(n + 1)`

`-\int u^(3)du = -[(u^(4))/(4) ]`

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

`= -[(u^(4))/(4) ] + C`

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

`= -(cos^(4)(x))/(4) + C`

And you're done!