Solve ∫ (from 0 to 1) of xe^-x4 wrt x

$$\int_{ 0 }^{ 1 } x { e }^{ -x4 } d x$$

Evaluate

$$\frac{1}{2e^{x_{4}}}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{x}{e^{x_{4}}}\mathrm{d}x$$

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

$$\frac{\int x\mathrm{d}x}{e^{x_{4}}}$$

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}.

$$\frac{x^{2}}{2e^{x_{4}}}$$

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.

$$\frac{1}{2}e^{-x_{4}}\times 1^{2}-\frac{1}{2}e^{-x_{4}}\times 0^{2}$$

Simplify.

$$\frac{1}{2e^{x_{4}}}$$

Differentiate w.r.t. x_4

$$-\frac{1}{2e^{x_{4}}}$$