"Find the indefinite integral. ∫x2x−4dx" solved and explained

Betsy Rhone

Answered question

2021-12-14

Find the indefinite integral.

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

jean2098

Beginner2021-12-15Added 38 answers

Step 1: Given that:

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

We need to solve the given integral.
Step 2: Solving the given integral

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

On adding and subtracting 16 in the numerator of the integral we get,

\int \frac{x^{2}}{x - 4} d x = \int \frac{x^{2} + 16 - 16}{x - 4} d x

\Rightarrow \int \frac{x^{2}}{x - 4} d x = \int \frac{x^{2} - 16}{x - 4} d x + \int \frac{16}{x - 4} d x

\Rightarrow \int \frac{x^{2}}{x - 4} d x = \int \frac{x^{2} - 4^{2}}{x - 4} d x + \int \frac{16}{x - 4} d x

\Rightarrow \int \frac{x^{2}}{x - 4} d x = \int \frac{(x - 4) (x + 4)}{x - 4} d x + \int \frac{16}{x - 4} d x

\Rightarrow \int \frac{x^{2}}{x - 4} d x = \int (x + 4) d x + \int \frac{16}{x - 4} d x

\Rightarrow \int \frac{x^{2}}{x - 4} d x = \int x d x + \int 4 \cdot d x + 16 \int \frac{1}{x - 4} d x

\Rightarrow \int \frac{x^{2}}{x - 4} d x = \frac{x^{2}}{2} + 4 x + 16 \ln | x - 4 | + C

Neil Dismukes

Beginner2021-12-16Added 37 answers

Step 1

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

Using the long division

\frac{x^{2}}{x - 4} = x + 4 + \frac{x}{x - 4}

, so

\int \frac{x^{2}}{x - 4} d x = \int (x + 4 + \frac{16}{x - 4}) d x

Write as a sum of integrals

\int x d x + \int 4 + \int \frac{16}{x - 4} d x

Integrate

\frac{x^{2}}{2} + 4 x + 16 \ln | x - 4 | + C

Result:

\frac{x^{2}}{2} + 4 x + 16 \ln | x - 4 | + C

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