Solve ∫ (from 0 to pi) of (4e^x+4sin(x)) wrt x

$$\int _ { 0 } ^ { \pi } ( 4 e ^ { x } + 4 \sin ( x ) ) d x$$

Evaluate

$$4\left(e^{\pi }+1\right)\approx 96.562770531$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4\left(e^{x}+\sin(x)\right)\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4e^{x}\mathrm{d}x+\int 4\sin(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\left(\int e^{x}\mathrm{d}x+\int \sin(x)\mathrm{d}x\right)$$

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

$$4\left(e^{x}+\int \sin(x)\mathrm{d}x\right)$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 4 times -\cos(x).

$$4\left(e^{x}-\cos(x)\right)$$

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.

$$4e^{\pi }-4\cos(\pi )-\left(4e^{0}-4\cos(0)\right)$$

Simplify.

$$4e^{\pi }+4$$