Solve ∫ (from 0 to pi) of u^2sinx

$$\int _ { 0 } ^ { \pi } u ^ { 2 } \sin x$$

Evaluate

$$2u^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int u^{2}\sin(x)\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$u^{2}\int \sin(x)\mathrm{d}x$$

Use \int \sin(u)\mathrm{d}u=-\cos(u) from the table of common integrals to obtain the result.

$$-u^{2}\cos(x)$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$-u^{2}\cos(\pi )+u^{2}\cos(0)$$

Simplify.

$$2u^{2}$$

Differentiate w.r.t. u

$$4u$$