What is the integral of ((x^3))/(sqrt(x^2+81)) ?

The integral of ((x^3))/(sqrt(x^2+81)) is 1/3 (x^2+81)^{3/2}-81sqrt(x^2+81)+C

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integral of ((x^3))/(sqrt(x^2+81))

\int \frac{( x ^{3} )}{x ^{2} + 81} d x

\frac{1}{3} (x^{2} + 81)^{\frac{3}{2}} - 81 x^{2} + 81 + C

Solution steps

\int \frac{x ^{3}}{x ^{2} + 81} d x

Apply u-substitution

: \int u^{2} - 81 d u

= \int u^{2} - 81 d u

Apply the Sum Rule:

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

= \int u^{2} d u - \int 81 d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 81 d u = 81 u

= \frac{u ^{3}}{3} - 81 u

Substitute back

u = x^{2} + 81

= \frac{( x ^{2} + 81 ) ^{3}}{3} - 81 x^{2} + 81

Simplify

\frac{( x ^{2} + 81 ) ^{3}}{3} - 81 x^{2} + 81 : \frac{1}{3} (x^{2} + 81)^{\frac{3}{2}} - 81 x^{2} + 81

= \frac{1}{3} (x^{2} + 81)^{\frac{3}{2}} - 81 x^{2} + 81

Add a constant to the solution

= \frac{1}{3} (x^{2} + 81)^{\frac{3}{2}} - 81 x^{2} + 81 + C

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What is the integral of ((x^3))/(sqrt(x^2+81)) ?
The integral of ((x^3))/(sqrt(x^2+81)) is 1/3 (x^2+81)^{3/2}-81sqrt(x^2+81)+C