Q. 18 - Calculus

Question

Indefinite integrals: Use integration formulas, algebra, and

educated guess-and-check strategies to find the following

integrals .

$\int \frac{1-x+x^7}{x^2} dx$

Step-by-Step Solution

Answer

The integral is $\left(-\frac{1}{x}-In\left|x\right|+\frac{x^6}{6}+C\right)$ .

Step 1. Given information .

Consider the given integral $\int \frac{1-x+x^7}{x^2} dx$ .

Step 2. Using the sum rule .

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

Step 3. Find the integral .

$$\begin{gathered} \Rightarrow \int \frac{1-x+x^7}{x^2} dx \\ \Rightarrow \int \frac{1}{x^2}-\frac{1}{x}+x^5 dx \\ \Rightarrow \int \frac{1}{x^2}dx-\int \frac{1}{x}dx+\int x^{5}dx \\ \Rightarrow \frac{-1}{x}-In \left|x\right|+\frac{x^6}{6}+C \end{gathered}$$