Q. 23 - Calculus

Question

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.  

$\int \left(2x{3}^x+x^2\left(\ln 3\right)3^x\right)dx$

Step-by-Step Solution

Answer

The solution is: $x^2{3}^x+C$

Step 1: Given

The given integral is: $\int \left(2x{3}^x+x^2\left(\ln 3\right)3^x\right)dx$

Step 2: To Find

Use integration formulas, algebra, and educated guess-and-check strategies to find the given integral.

Step 3: Calculation

Let,

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

Using substitution we get:

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

Where, C = integrating constant.