Q. 19 - Calculus

Question

Indefinite integrals: Use integration formulas, algebra, and

educated guess-and-check strategies to find the following

integrals .

$\int e^x\left(1-e^{3x}\right) dx$

Step-by-Step Solution

Answer

The integral is $\left(e^x-\frac{1}{4}{e}^{4x}+C\right)$ .

Step 1. Given information .

Consider the given integral $\int e^x\left(1-e^{3x}\right) dx$ .

Step 2. Using the sum rule .

The sum rule is $\int f\left(x\right)\pm g\left(x\right) dx=\int f\left(x\right) dx \pm \int g\left(x\right) dx$ .

Step 3. Find the integral .

$$\begin{gathered} \Rightarrow \int e^x\left(1-e^{3x}\right) dx \\ \Rightarrow \int e^x dx-\int e^{4x} dx \\ \Rightarrow e^x-\frac{1}{4}{e}^{4x}+C \end{gathered}$$