What is the integral of (5/3 x^3sqrt(x^4-3)) ?

The integral of (5/3 x^3sqrt(x^4-3)) is 5/18 (x^4-3)^{3/2}+C

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integral of (5/3 x^3sqrt(x^4-3))

\int (\frac{5}{3} x^{3} x^{4} - 3) d x

\frac{5}{18} (x^{4} - 3)^{\frac{3}{2}} + C

Solution steps

\int \frac{5}{3} x^{3} x^{4} - 3 d x

Take the constant out:

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

= \frac{5}{3} \cdot \int x^{3} x^{4} - 3 d x

Apply u-substitution

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

= \frac{5}{3} \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

= \frac{5}{3} \cdot \frac{1}{4} \cdot \int u d u

Apply the Power Rule

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

= \frac{5}{3} \cdot \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x^{4} - 3

= \frac{5}{3} \cdot \frac{1}{4} \cdot \frac{2}{3} (x^{4} - 3)^{\frac{3}{2}}

Simplify

\frac{5}{3} \cdot \frac{1}{4} \cdot \frac{2}{3} (x^{4} - 3)^{\frac{3}{2}} : \frac{5}{18} (x^{4} - 3)^{\frac{3}{2}}

= \frac{5}{18} (x^{4} - 3)^{\frac{3}{2}}

Add a constant to the solution

= \frac{5}{18} (x^{4} - 3)^{\frac{3}{2}} + C

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\int \frac{1}{x ^{4} + 3 x ^{2}} d x

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\int 3 e^{0.5 x} d x

\int \frac{1}{x ( x + b )} d x

What is the integral of (5/3 x^3sqrt(x^4-3)) ?
The integral of (5/3 x^3sqrt(x^4-3)) is 5/18 (x^4-3)^{3/2}+C