Solve ∫ (from - 3 to 1) of 4cos(x)+3e^x wrt x

$$\int _ { - 3 } ^ { 1 } 4 \cos ( x ) + 3 e ^ { x } d x$$

Evaluate

$$-\frac{3}{e^{3}}+3e+4\sin(1)+4\sin(3)\approx 11.935848252$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

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

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.

$$4\sin(1)+3e^{1}-\left(4\sin(-3)+3e^{-3}\right)$$

Simplify.

$$3e-\frac{3}{e^{3}}+4\sin(1)+4\sin(3)$$