What is the integral of 7x^3sqrt(x^4+6) ?

The integral of 7x^3sqrt(x^4+6) is 7/6 (x^4+6)^{3/2}+C

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integral of 7x^3sqrt(x^4+6)

\int 7 x^{3} x^{4} + 6 d x

\frac{7}{6} (x^{4} + 6)^{\frac{3}{2}} + C

Solution steps

\int 7 x^{3} x^{4} + 6 d x

Take the constant out:

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

= 7 \cdot \int x^{3} x^{4} + 6 d x

Apply u-substitution

: \int \frac{u}{4} d u

= 7 \cdot \int \frac{u}{4} d u

Take the constant out:

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

= 7 \cdot \frac{1}{4} \cdot \int u d u

Apply the Power Rule

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

= 7 \cdot \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x^{4} + 6

= 7 \cdot \frac{1}{4} \cdot \frac{2}{3} (x^{4} + 6)^{\frac{3}{2}}

Simplify

7 \cdot \frac{1}{4} \cdot \frac{2}{3} (x^{4} + 6)^{\frac{3}{2}} : \frac{7}{6} (x^{4} + 6)^{\frac{3}{2}}

= \frac{7}{6} (x^{4} + 6)^{\frac{3}{2}}

Add a constant to the solution

= \frac{7}{6} (x^{4} + 6)^{\frac{3}{2}} + C

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What is the integral of 7x^3sqrt(x^4+6) ?
The integral of 7x^3sqrt(x^4+6) is 7/6 (x^4+6)^{3/2}+C