Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$\int \left(x^2+2^x+2^2\right)dx$

Short answer

The answer is$\frac{x^3}{3}+\frac{2^x}{\ln \left(2\right)}+4x+C$.

Step-by-step solution

Step 1. Given Information.

The given integral is$\int \left(x^2+2^x+2^2\right)dx$.

Step 2. Integration.

On integrating, we get,

$$\begin{gathered} \int \left(x^2+2^x+2^2\right)dx \\ =\int x^{2}dx+\int 2^{x}dx+\int 4dx \\ =\frac{x^3}{3}+\frac{2^x}{\ln \left(2\right)}+4x+C \end{gathered}$$

Step 3. Verification.

On differentiating $\frac{x^3}{3}+\frac{2^x}{\ln \left(2\right)}+4x+C$, we get,

$$\begin{gathered} =\frac{3x^2}{3}+2^x+4 \\ =\frac{3x^2}{3}+2^x+2^2 \end{gathered}$$

Hence proved.