What is the Integral of sin^3(x)cos^2(x)?

[Jumpy Lee]

Homework Statement

$$\int sin^3(x)cos^2(x)dx$$

The Attempt at a Solution

$$\int sin^3x(1-sin^2x)dx =$$ $$\int sin^3x- sin^5xdx$$

i get stuck here what do i do next and is this even right

 

[niceperson]

we could use complex numbers (de moivres theorem) here and express sin^5x, sin^3x in terms of sin5x, sin3x

(directly integratable)

 

[Jumpy Lee]

is that from calc 2?

 

[dextercioby]

Homework Statement

$$\int sin^3(x)cos^2(x)dx$$

The Attempt at a Solution

$$\int sin^3x(1-sin^2x)dx =$$ $$\int sin^3x- sin^5xdx$$

i get stuck here what do i do next and is this even right

HINT

$$\int \left(1-\cos^{2}x\right) \cos^{2}x \left(\sin x \ dx\right)$$

Do you know how to use the substitution method ?

 

[Jumpy Lee]

yes i do know how to use substution

 

[dextercioby]

Do you see a possible substitution in what i wrote ?

 

[Jumpy Lee]

u = cos x -du = sin x dx right?

 

[dextercioby]

Okay. Very good. Now use it.

 

[Jumpy Lee]

-$$\int (u^2 - u^4)du =$$ -u^3/3 + u^5/5 +c = -cos^3x/3 + cos^5x/5 + c

 

[dextercioby]

Well done.