Solve ∫ (from 0 to t) of sinhue^-2(u-t)

$$\int _ { 0 } ^ { t } \sinh u e ^ { - 2 ( u - t ) }$$

Evaluate

$$t\sinh(ue^{2\left(t-u\right)})$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sinh(\frac{u}{e^{2\left(u-t\right)}})\mathrm{d}x$$

Find the integral of \sinh(\frac{u}{e^{2\left(u-t\right)}}) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\sinh(\frac{u}{e^{2\left(u-t\right)}})x$$

Simplify.

$$\sinh(ue^{-2u+2t})x$$

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.

$$\sinh(ue^{-2u+2t})t-\sinh(ue^{-2u+2t})\times 0$$

Simplify.

$$\sinh(ue^{-2u+2t})t$$

Differentiate w.r.t. u

$$t\left(1-2u\right)e^{2\left(t-u\right)}\cosh(ue^{2\left(t-u\right)})$$