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

The integral of x^3sqrt(1+(3x^2)^2) is 1/54 (1+9x^4)^{3/2}+C

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

\int x^{3} 1 + (3 x^{2})^{2} d x

\frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}} + C

Solution steps

\int x^{3} 1 + (3 x^{2})^{2} d x

Apply u-substitution

: \int \frac{u}{36} d u

= \int \frac{u}{36} d u

Take the constant out:

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

= \frac{1}{36} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{36} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + (3 x^{2})^{2}

= \frac{1}{36} \cdot \frac{2}{3} (1 + (3 x^{2})^{2})^{\frac{3}{2}}

Simplify

\frac{1}{36} \cdot \frac{2}{3} (1 + (3 x^{2})^{2})^{\frac{3}{2}} : \frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}}

= \frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}} + C

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What is the integral of x^3sqrt(1+(3x^2)^2) ?
The integral of x^3sqrt(1+(3x^2)^2) is 1/54 (1+9x^4)^{3/2}+C