What is the integral of 30x^3sqrt(1+x^2) ?

The integral of 30x^3sqrt(1+x^2) is 15(2/5 (x^2+1)^{5/2}-2/3 (x^2+1)^{3/2})+C

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integral of 30x^3sqrt(1+x^2)

\int 30 x^{3} 1 + x^{2} d x

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

Solution steps

\int 30 x^{3} 1 + x^{2} d x

Take the constant out:

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

= 30 \cdot \int x^{3} 1 + x^{2} d x

Apply u-substitution

= 30 \cdot \int \frac{( u - 1 ) u}{2} d u

Take the constant out:

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

= 30 \cdot \frac{1}{2} \cdot \int (u - 1) u d u

Expand

(u - 1) u : u^{\frac{3}{2}} - u

= 30 \cdot \frac{1}{2} \cdot \int u^{\frac{3}{2}} - u 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

= 30 \cdot \frac{1}{2} (\int u^{\frac{3}{2}} d u - \int u d u)

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

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

= 30 \cdot \frac{1}{2} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 1 + x^{2}

= 30 \cdot \frac{1}{2} (\frac{2}{5} (1 + x^{2})^{\frac{5}{2}} - \frac{2}{3} (1 + x^{2})^{\frac{3}{2}})

Simplify

30 \cdot \frac{1}{2} (\frac{2}{5} (1 + x^{2})^{\frac{5}{2}} - \frac{2}{3} (1 + x^{2})^{\frac{3}{2}}) : 15 (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}})

= 15 (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}})

Add a constant to the solution

= 15 (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}) + C

What is the integral of 30x^3sqrt(1+x^2) ?
The integral of 30x^3sqrt(1+x^2) is 15(2/5 (x^2+1)^{5/2}-2/3 (x^2+1)^{3/2})+C