Solve ∫ (3e^x+2sinx) wrt x

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

Evaluate

$$3e^{x}-2\cos(x)+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 2 times -\cos(x).

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

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$3e^{x}-2\cos(x)+С$$

Differentiate w.r.t. x

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