"Find the indefinite integral. ∫x3−4x2−4x+20x2−5dx" was answered by our experts

Michelle Arroyo

Answered question

2021-11-18

Find the indefinite integral.

\int \frac{x^{3} - 4 x^{2} - 4 x + 20}{x^{2} - 5} d x

David Tyson

Beginner2021-11-19Added 19 answers

Step 1
find the indefinite integral.

\int \frac{x^{3} - 4 x^{2} - 4 x + 20}{x^{2} - 5} d x

Step 2

I = \int \frac{x^{3} - 4 x^{2} - 4 x + 20}{x^{2} - 5} d x

= \int (\frac{x}{x^{2} - 5} + x - 4) d x

= \int \frac{x}{x^{2} - 5} d x + \int x d x - \int 4 d x

= \frac{1}{2} \log (x^{2} - 5) + \frac{x^{2}}{2} - 4 x + c

Ruth Phillips

Beginner2021-11-20Added 18 answers

Step 1: Polynomial Division

\int x - 4 + \frac{x}{x^{2} - 5} d x

Step 2: Use Sum Rule:

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

\int x - 4 d x + \int \frac{x}{x^{2} - 5} d x

Step 3: Use Power Rule:

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

\frac{x^{2}}{2} - 4 x + \int \frac{x}{x^{2} - 5} d x

Step 4: Use Integration by Substitution on

\int \frac{x}{x^{2} - 5} d x

.
Let

u = x^{2} - 5, d u = 2 x d x, t h e n x d x = \frac{1}{2} d u

Step 5: Using u and du above, rewrite

\int \frac{x}{x^{2} - 5} d x

\int \frac{1}{2 u} d u

Step 6: Use Constant Factor Rule:

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

\frac{1}{2} \int \frac{1}{u} d u

Step 7: The derivative of

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

\frac{\ln u}{2}

Step 8: Substitute

u = x^{2} - 5

back into the original integral.

\frac{\ln (x^{2} - 5)}{2}

Step 9: Rewrite the integral with the completed substitution.

\frac{x^{2}}{2} - 4 x + \frac{\ln (x^{2} - 5)}{2}

Step 10: Add constant.

\frac{x^{2}}{2} - 4 x + \frac{\ln (x^{2} - 5)}{2} + C

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