What is the integral of (2x^3sqrt(x^2-5)) ?

The integral of (2x^3sqrt(x^2-5)) is 2(1/5 (x^2-5)^{5/2}+5/3 (x^2-5)^{3/2})+C

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integral of (2x^3sqrt(x^2-5))

\int (2 x^{3} x^{2} - 5) d x

2 (\frac{1}{5} (x^{2} - 5)^{\frac{5}{2}} + \frac{5}{3} (x^{2} - 5)^{\frac{3}{2}}) + C

Solution steps

\int 2 x^{3} x^{2} - 5 d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x^{3} x^{2} - 5 d x

Apply u-substitution

: \int (u^{2} + 5) u^{2} d u

= 2 \cdot \int (u^{2} + 5) u^{2} d u

Expand

(u^{2} + 5) u^{2} : u^{4} + 5 u^{2}

= 2 \cdot \int u^{4} + 5 u^{2} 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

= 2 (\int u^{4} d u + \int 5 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 5 u^{2} d u = \frac{5 u ^{3}}{3}

= 2 (\frac{u ^{5}}{5} + \frac{5 u ^{3}}{3})

Substitute back

u = x^{2} - 5

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

Simplify

2 (\frac{( x ^{2} - 5 ) ^{5}}{5} + \frac{5 ( x ^{2} - 5 ) ^{3}}{3}) : 2 (\frac{1}{5} (x^{2} - 5)^{\frac{5}{2}} + \frac{5}{3} (x^{2} - 5)^{\frac{3}{2}})

= 2 (\frac{1}{5} (x^{2} - 5)^{\frac{5}{2}} + \frac{5}{3} (x^{2} - 5)^{\frac{3}{2}})

Add a constant to the solution

= 2 (\frac{1}{5} (x^{2} - 5)^{\frac{5}{2}} + \frac{5}{3} (x^{2} - 5)^{\frac{3}{2}}) + C

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What is the integral of (2x^3sqrt(x^2-5)) ?
The integral of (2x^3sqrt(x^2-5)) is 2(1/5 (x^2-5)^{5/2}+5/3 (x^2-5)^{3/2})+C