What is the integral of (x^2)/((x^2-9)^{5/2)} ?

The integral of (x^2)/((x^2-9)^{5/2)} is (x^3)/(3(x^2-9)^{5/2)}-(x^5)/(27(x^2-9)^{5/2)}+C

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integral of (x^2)/((x^2-9)^{5/2)}

\int \frac{x ^{2}}{( x ^{2} - 9 ) ^{\frac{5}{2}}} d x

\frac{x ^{3}}{3 ( x ^{2} - 9 ) ^{\frac{5}{2}}} - \frac{x ^{5}}{27 ( x ^{2} - 9 ) ^{\frac{5}{2}}} + C

Solution steps

\int \frac{x ^{2}}{( x ^{2} - 9 ) ^{\frac{5}{2}}} d x

Apply Integration By Parts

: \frac{x ^{3}}{3 ( x ^{2} - 9 ) ^{\frac{5}{2}}} - \int - \frac{5 x ^{4}}{3 ( x ^{2} - 9 ) ^{\frac{7}{2}}} d x

= \frac{x ^{3}}{3 ( x ^{2} - 9 ) ^{\frac{5}{2}}} - \int - \frac{5 x ^{4}}{3 ( x ^{2} - 9 ) ^{\frac{7}{2}}} d x

\int - \frac{5 x ^{4}}{3 ( x ^{2} - 9 ) ^{\frac{7}{2}}} d x = \frac{x ^{5}}{27 ( x ^{2} - 9 ) ^{\frac{5}{2}}}

= \frac{x ^{3}}{3 ( x ^{2} - 9 ) ^{\frac{5}{2}}} - \frac{x ^{5}}{27 ( x ^{2} - 9 ) ^{\frac{5}{2}}}

Add a constant to the solution

= \frac{x ^{3}}{3 ( x ^{2} - 9 ) ^{\frac{5}{2}}} - \frac{x ^{5}}{27 ( x ^{2} - 9 ) ^{\frac{5}{2}}} + C

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What is the integral of (x^2)/((x^2-9)^{5/2)} ?
The integral of (x^2)/((x^2-9)^{5/2)} is (x^3)/(3(x^2-9)^{5/2)}-(x^5)/(27(x^2-9)^{5/2)}+C