Solve ∫ (from - pi to pi) of (3sinx+4cosx+2e^x) wrt x

$$\int _ { - \pi } ^ { \pi } ( 3 \sin x + 4 \cos x + 2 e ^ { x } ) d x$$

Evaluate

$$-\frac{2}{e^{\pi }}+2e^{\pi }\approx 46.194957429$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3\sin(x)+4\cos(x)+2e^{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3\sin(x)\mathrm{d}x+\int 4\cos(x)\mathrm{d}x+\int 2e^{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int \sin(x)\mathrm{d}x+4\int \cos(x)\mathrm{d}x+2\int e^{x}\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 3 times -\cos(x).

$$-3\cos(x)+4\int \cos(x)\mathrm{d}x+2\int e^{x}\mathrm{d}x$$

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

$$-3\cos(x)+4\sin(x)+2\int e^{x}\mathrm{d}x$$

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

$$-3\cos(x)+4\sin(x)+2e^{x}$$

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.

$$-3\cos(\pi )+4\sin(\pi )+2e^{\pi }-\left(-3\cos(-\pi )+4\sin(-\pi )+2e^{-\pi }\right)$$

Simplify.

$$2e^{\pi }-\frac{2}{e^{\pi }}$$