"Evaluate the integral. ∫(1−xx)2dx" was answered by our experts

chanyingsauu7

Answered question

2021-11-16

Evaluate the integral.

\int {(\frac{1 - x}{x})}^{2} d x

Mike Henson

Beginner2021-11-17Added 11 answers

Step 1: Given that
Evaluate the integral.

\int {(\frac{1 - x}{x})}^{2} d x

Step 2: Solve
We have,

\int {(\frac{1 - x}{x})}^{2} d x = \int \frac{1 - 2 x + x^{2}}{x^{2}} d x

= \int \frac{1}{x^{2}} d x - 2 \int \frac{x}{x^{2}} + \int \frac{x^{2}}{x^{2}} d x

= \int x^{- 2} d x - 2 \int \frac{1}{x} d x + \int 1 d x

= \frac{x^{- 2 + 1}}{- 2 + 1} - 2 \ln | x | + x + C

= - \frac{1}{x} - 2 \ln | x | + x + C

= x - 2 \ln | x | - \frac{1}{x} + C

Nancy Johnson

Beginner2021-11-18Added 17 answers

Step 1: Simplify

\frac{1 - x}{x} \to - 1 + \frac{1}{x}

\int {(- 1 + \frac{1}{x})}^{2} d x

Step 2: Regroup terms.

\int {(\frac{1}{x} - 1)}^{2} d x

Step 3: Expand.

\int \frac{1}{x^{2}} - \frac{2}{x} + 1 d x

Step 4: Use Sum Rule:

\int f (x) + g (x) d x = \int f (x) d x + \int g (x) d x

\int \frac{1}{x^{2}} d x - \int \frac{2}{x} d x + \int 1 d x

Step 5: Use Power Rule:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

- \frac{1}{x} - \int \frac{2}{x} d x + \int 1 d x

Step 6: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

- \frac{1}{x} - 2 \int \frac{1}{x} d x + \int 1 d x

Step 7: The derivative of

\ln x i s \frac{1}{x}

- \frac{1}{x} - 2 \ln x + \int 1 d x

Step 8: Use this rule:

\int a d x = a x + C

- \frac{1}{x} - 2 \ln x + x

Step 9: Add constant.

- \frac{1}{x} - 2 \ln x + x + C

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