Solve ∫ (from 2 to 5) of e^x+5x wrt x

$$\int_{ 2 }^{ 5 } { e }^{ x } +5x d x$$

Evaluate

$$e^{5}-e^{2}+\frac{105}{2}\approx 193.524103004$$

Solution Steps

Evaluate the indefinite integral first.

$$\int e^{x}+5x\mathrm{d}x$$

Integrate the sum term by term.

$$\int e^{x}\mathrm{d}x+\int 5x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int e^{x}\mathrm{d}x+5\int x\mathrm{d}x$$

Use \int e^{x}\mathrm{d}x=e^{x} from the table of common integrals to obtain the result.

$$e^{x}+5\int x\mathrm{d}x$$

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}. Multiply 5 times \frac{x^{2}}{2}.

$$e^{x}+\frac{5x^{2}}{2}$$

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.

$$e^{5}+\frac{5}{2}\times 5^{2}-\left(e^{2}+\frac{5}{2}\times 2^{2}\right)$$

Simplify.

$$e^{5}+\frac{105}{2}-e^{2}$$