Solve ∫ (from 0 to t) of usin(t-mu)du

$$\int _ { 0 } ^ { t } u \sin ( t - \mu ) d u$$

Evaluate

$$\frac{t^{2}\sin(t-\mu )}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int u\sin(t-\mu )\mathrm{d}u$$

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

$$\sin(t-\mu )\int u\mathrm{d}u$$

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

$$\sin(t-\mu )\times \frac{u^{2}}{2}$$

Simplify.

$$\frac{\sin(t-\mu )u^{2}}{2}$$

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.

$$\frac{1}{2}\sin(t-\mu )t^{2}-\frac{1}{2}\sin(t-\mu )\times 0^{2}$$

Simplify.

$$\frac{\sin(t-\mu )t^{2}}{2}$$

Differentiate w.r.t. t

$$\frac{t\left(t\cos(t-\mu )+2\sin(t-\mu )\right)}{2}$$