Solve ∫ x^2exp(x) wrt x

$$\int x ^ { 2 } \exp ( x ) d x$$

Evaluate

$$\frac{epx^{5}}{5}+С$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\int x^{3}epx\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\int x^{4}ep\mathrm{d}x$$

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

$$ep\int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}.

$$ep\times \frac{x^{5}}{5}$$

Simplify.

$$\frac{epx^{5}}{5}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{epx^{5}}{5}+С$$

Differentiate w.r.t. x

$$epx^{4}$$