Question

### Gauthmathier3143

Grade 11 · 2021-01-10

\int \dfrac {x\d x}{x^{2}-3x-4}

Good Question (94)

Answer

4.9(151) votes

### Gauthmathier5260

Grade 11 · 2021-01-10

Answer

\dfrac {1}{5}\ln |(x-4)^{4}(x+1)|+C.

Explanation

Factor the denominator so that \int \dfrac {x\d x}{x^{2}-3x-4}=\int \dfrac {x\d x}{(x-4)(x+1)}.

Use a partial fraction decomposition, \dfrac {x}{(x-4)(x+1)}=\dfrac {A}{(x-4)}+\dfrac {B}{(x+1)}, which implies Ax+A+Bx-4B=x. Solve A+B=1 and A-4B to find A=\dfrac {4}{5} and B=\dfrac {1}{5}. Integrate \int \dfrac {x\d x}{x^{2}-3x-4}=\int \frac {\frac{4}{5}}{x-4}\d x+\int \frac {\frac{1}{5}}{x+1}\d x=\dfrac {4}{5}\ln |x-4|+\dfrac {1}{5}\ln |x+1|+C=\dfrac {1}{5}\ln |(x-4)^{4}(x+1)|+C.

Use a partial fraction decomposition, \dfrac {x}{(x-4)(x+1)}=\dfrac {A}{(x-4)}+\dfrac {B}{(x+1)}, which implies Ax+A+Bx-4B=x. Solve A+B=1 and A-4B to find A=\dfrac {4}{5} and B=\dfrac {1}{5}. Integrate \int \dfrac {x\d x}{x^{2}-3x-4}=\int \frac {\frac{4}{5}}{x-4}\d x+\int \frac {\frac{1}{5}}{x+1}\d x=\dfrac {4}{5}\ln |x-4|+\dfrac {1}{5}\ln |x+1|+C=\dfrac {1}{5}\ln |(x-4)^{4}(x+1)|+C.

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