Solve ∫ (from 0 to 3) of exp(x^2) wrt x

$$\int _ { 0 } ^ { 3 } \exp ( x ^ { 2 } ) d x$$

Evaluate

$$\frac{81ep}{4}$$

Solution Steps

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

$$\int _{0}^{3}ex^{3}p\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int ex^{3}p\mathrm{d}x$$

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

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$ep\times \frac{x^{4}}{4}$$

Simplify.

$$\frac{epx^{4}}{4}$$

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.

$$\frac{1}{4}ep\times 3^{4}-\frac{1}{4}ep\times 0^{4}$$

Simplify.

$$\frac{81ep}{4}$$

Differentiate w.r.t. p

$$\frac{81e}{4}$$