What is the integral of (x^3)/(sqrt(x^2-49)) ?

The integral of (x^3)/(sqrt(x^2-49)) is 1/3 (x^2-49)^{3/2}+49sqrt(x^2-49)+C

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integral of (x^3)/(sqrt(x^2-49))

\int \frac{x ^{3}}{x ^{2} - 49} d x

\frac{1}{3} (x^{2} - 49)^{\frac{3}{2}} + 49 x^{2} - 49 + C

Solution steps

\int \frac{x ^{3}}{x ^{2} - 49} d x

Apply u-substitution

: \int u^{2} + 49 d u

= \int u^{2} + 49 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 49 d u

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

\int 49 d u = 49 u

= \frac{u ^{3}}{3} + 49 u

Substitute back

u = x^{2} - 49

= \frac{( x ^{2} - 49 ) ^{3}}{3} + 49 x^{2} - 49

Simplify

\frac{( x ^{2} - 49 ) ^{3}}{3} + 49 x^{2} - 49 : \frac{1}{3} (x^{2} - 49)^{\frac{3}{2}} + 49 x^{2} - 49

= \frac{1}{3} (x^{2} - 49)^{\frac{3}{2}} + 49 x^{2} - 49

Add a constant to the solution

= \frac{1}{3} (x^{2} - 49)^{\frac{3}{2}} + 49 x^{2} - 49 + C

\frac{d}{d x} (1 - e^{- x^{2}})

x \to 2 lim (x^{4} + 5^{x} - 2)

x \to - 1 lim (2 x^{3} + 3 x^{2} - 6)

\int_{- 1}^{3} x^{2} - 2 x d x

\frac{1}{x ^{\frac{1}{2}}}

What is the integral of (x^3)/(sqrt(x^2-49)) ?
The integral of (x^3)/(sqrt(x^2-49)) is 1/3 (x^2-49)^{3/2}+49sqrt(x^2-49)+C