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

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

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integral of 9x^3sqrt(3x^4+4)

\int 9 x^{3} 3 x^{4} + 4 d x

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

Solution steps

\int 9 x^{3} 3 x^{4} + 4 d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u}{12} d u

= 9 \cdot \int \frac{u}{12} d u

Take the constant out:

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

= 9 \cdot \frac{1}{12} \cdot \int u d u

Apply the Power Rule

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

= 9 \cdot \frac{1}{12} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 3 x^{4} + 4

= 9 \cdot \frac{1}{12} \cdot \frac{2}{3} (3 x^{4} + 4)^{\frac{3}{2}}

Simplify

9 \cdot \frac{1}{12} \cdot \frac{2}{3} (3 x^{4} + 4)^{\frac{3}{2}} : \frac{1}{2} (3 x^{4} + 4)^{\frac{3}{2}}

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

Add a constant to the solution

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

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