Question

Use substitution and do integration x^3sin(x^2)dx

Answer 1

∫ x^3 sin(x^2)dx = {1} / {2}(-x^2 cos(x^2) + sin(x^2)) + C

Step-by-step explanation:

To solve the integral (∫ x^3 sin(x^2),dx) using substitution, let u = x^2, so du/dx = 2x.

Now, differentiate both sides with respect to x:

du/dx = 2x ⟹ du = 2x,dx

Rearrange the equation to solve for dx:

dx = du/(2x).

Now substitute (u) and dx into the original integral:

∫ x^3 sin(x^2),dx = ∫ x^2 . x sin(x^2),dx.

Substitute (u = x^2) and (dx = du/(2x)):

= ∫ u sin(u),du(2x).

Now, the integral becomes:

= (1/2) ∫ u sin(u),du.

Now integrate with respect to u:

= (1/2) (-u cos(u) + ∫ cos(u)du).

= (1/2) (-u cos(u) + sin(u)) + C,

where (C) is the constant of integration.

Now substitute back (u = x^2):

= (1/2) (-x^2 cos(x^2) + sin(x^2)) + C.

So, (∫ x^3 sin(x^2),dx = (1/2) (-x^2 cos(x^2) + sin(x^2)) + C), where C is the constant of integration.