Solve ∫ (from 0 to cos y) of xsiny wrt x

$$\int _ { 0 } ^ { \cos y } x \sin y d x$$

Evaluate

$$\frac{\sin(y)+\sin(3y)}{8}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x\sin(y)\mathrm{d}x$$

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

$$\sin(y)\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

$$\sin(y)\times \frac{x^{2}}{2}$$

Simplify.

$$\frac{\sin(y)x^{2}}{2}$$

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.

$$\frac{1}{2}\sin(y)\left(\cos(y)\right)^{2}-\frac{1}{2}\sin(y)\times 0^{2}$$

Simplify.

$$\frac{\sin(y)+\sin(3y)}{8}$$

Differentiate w.r.t. y

$$\frac{\cos(y)+3\cos(3y)}{8}$$