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

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

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

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

2 x^{3} + 1 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{3 u} d u

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

Take the constant out:

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

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

Apply the Power Rule

: 2 u^{\frac{1}{2}}

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

Substitute back

u = x^{3} + 1

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

Simplify

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

= 2 x^{3} + 1

Add a constant to the solution

= 2 x^{3} + 1 + C

h \to 0 lim (\frac{( h - 3 ) ^{2} - 9}{h})

8 + x^{2}, 1

f (x) = 4 x^{5} ln (x)

ln (1 - x^{2})

\int t^{2} e^{\frac{t}{3}} d t

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