Solve ∫ (3x^2-2^x+e^2) wrt x=

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

Evaluate

$$-\frac{2^{x}}{\ln(2)}+x^{3}+e^{2}x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 3x^{2}\mathrm{d}x+\int -2^{x}\mathrm{d}x+\int e^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{2}\mathrm{d}x-\int 2^{x}\mathrm{d}x+\int e^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 3 times \frac{x^{3}}{3}.

$$x^{3}-\int 2^{x}\mathrm{d}x+\int e^{2}\mathrm{d}x$$

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

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

Find the integral of e^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{3}-\frac{2^{x}}{\ln(2)}+e^{2}x$$

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.

$$x^{3}-\frac{2^{x}}{\ln(2)}+e^{2}x+С$$

Differentiate w.r.t. x

$$3x^{2}-2^{x}+e^{2}$$