Solve ∫ (2^x+1) wrt x

$$\int ( 2 ^ { x } + 1 ) d x$$

Evaluate

$$\frac{2^{x}}{\ln(2)}+x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2^{x}\mathrm{d}x+\int 1\mathrm{d}x$$

Use \int a^{b}\mathrm{d}b=\frac{a^{b}}{\ln(a)} from the table of common integrals to obtain the result.

$$\frac{2^{x}}{\ln(2)}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{2^{x}}{\ln(2)}+x$$

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.

$$\frac{2^{x}}{\ln(2)}+x+С$$

Differentiate w.r.t. x

$$2^{x}+1$$