Solve ∫ 2x^3ex^4 wrt x

$$\int{ 2 { x }^{ 3 } e { x }^{ 4 } }d x$$

Evaluate

$$\frac{ex^{8}}{4}+С$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$\int 2x^{7}e\mathrm{d}x$$

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

$$2e\int x^{7}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{7}\mathrm{d}x with \frac{x^{8}}{8}.

$$2e\times \frac{x^{8}}{8}$$

Simplify.

$$\frac{ex^{8}}{4}$$

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.

$$\frac{ex^{8}}{4}+С$$

Differentiate w.r.t. x

$$2ex^{7}$$