What is the integral of (3x^2)/(sqrt(5x^3+1)) ?

The integral of (3x^2)/(sqrt(5x^3+1)) is 2/5 sqrt(5x^3+1)+C

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integral of (3x^2)/(sqrt(5x^3+1))

\int \frac{3 x ^{2}}{5 x ^{3} + 1} d x

\frac{2}{5} 5 x^{3} + 1 + C

Solution steps

\int \frac{3 x ^{2}}{5 x ^{3} + 1} d x

Take the constant out:

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

= 3 \cdot \int \frac{x ^{2}}{5 x ^{3} + 1} d x

Apply u-substitution

: \int \frac{2}{15} d u

= 3 \cdot \int \frac{2}{15} d u

Integral of a constant:

\int a d x = a x

= 3 \cdot \frac{2}{15} u

Substitute back

u = 5 x^{3} + 1

= 3 \cdot \frac{2}{15} 5 x^{3} + 1

Simplify

3 \cdot \frac{2}{15} 5 x^{3} + 1 : \frac{2}{5} 5 x^{3} + 1

= \frac{2}{5} 5 x^{3} + 1

Add a constant to the solution

= \frac{2}{5} 5 x^{3} + 1 + C

4 y^{^{''}} - y = x e^{\frac{x}{2}}

\int 5 x^{4} - 3 x^{2} + 7 d x

x \to - 2.4 lim (x)

y^{^{'}} = \frac{2 y}{3 x}, y (27) = 4

\int \frac{x}{( x + 4 ) ( 2 x - 1 )} d x

What is the integral of (3x^2)/(sqrt(5x^3+1)) ?
The integral of (3x^2)/(sqrt(5x^3+1)) is 2/5 sqrt(5x^3+1)+C