Question

Indefinite integrals: Use integration formulas, algebra, and

educated guess-and-check strategies to find the following

integrals .

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

Short answer

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

Step-by-step solution

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 .

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