What is the integral of 2pix^3sqrt(1+9x^4) ?

The integral of 2pix^3sqrt(1+9x^4) is pi/(27)(9x^4+1)^{3/2}+C

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integral of 2pix^3sqrt(1+9x^4)

\int 2 π x^{3} 1 + 9 x^{4} d x

\frac{π}{27} (9 x^{4} + 1)^{\frac{3}{2}} + C

Solution steps

\int 2 π x^{3} 1 + 9 x^{4} d x

Take the constant out:

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

= 2 π \cdot \int x^{3} 1 + 9 x^{4} d x

Apply u-substitution

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

= 2 π \cdot \int \frac{u}{36} d u

Take the constant out:

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

= 2 π \frac{1}{36} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = 1 + 9 x^{4}

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

Simplify

2 π \frac{1}{36} \cdot \frac{2}{3} (1 + 9 x^{4})^{\frac{3}{2}} : \frac{π}{27} (9 x^{4} + 1)^{\frac{3}{2}}

= \frac{π}{27} (9 x^{4} + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{π}{27} (9 x^{4} + 1)^{\frac{3}{2}} + C

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What is the integral of 2pix^3sqrt(1+9x^4) ?
The integral of 2pix^3sqrt(1+9x^4) is pi/(27)(9x^4+1)^{3/2}+C