What is the integral of (x^3)/(sqrt(x^2+144)) ?

The integral of (x^3)/(sqrt(x^2+144)) is 1/3 (x^2+144)^{3/2}-144sqrt(x^2+144)+C

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integral of (x^3)/(sqrt(x^2+144))

\int \frac{x ^{3}}{x ^{2} + 144} d x

\frac{1}{3} (x^{2} + 144)^{\frac{3}{2}} - 144 x^{2} + 144 + C

Solution steps

\int \frac{x ^{3}}{x ^{2} + 144} d x

Apply u-substitution

= \int u^{2} - 144 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 144 d u

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

\int 144 d u = 144 u

= \frac{u ^{3}}{3} - 144 u

Substitute back

u = x^{2} + 144

= \frac{( x ^{2} + 144 ) ^{3}}{3} - 144 x^{2} + 144

Simplify

\frac{( x ^{2} + 144 ) ^{3}}{3} - 144 x^{2} + 144 : \frac{1}{3} (x^{2} + 144)^{\frac{3}{2}} - 144 x^{2} + 144

= \frac{1}{3} (x^{2} + 144)^{\frac{3}{2}} - 144 x^{2} + 144

Add a constant to the solution

= \frac{1}{3} (x^{2} + 144)^{\frac{3}{2}} - 144 x^{2} + 144 + C

What is the integral of (x^3)/(sqrt(x^2+144)) ?
The integral of (x^3)/(sqrt(x^2+144)) is 1/3 (x^2+144)^{3/2}-144sqrt(x^2+144)+C