What is the integral of (12x^2)/(2x^3+1) ?

The integral of (12x^2)/(2x^3+1) is 2ln|2x^3+1|+C

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integral of (12x^2)/(2x^3+1)

\int \frac{12 x ^{2}}{2 x ^{3} + 1} d x

2 ln 2 x^{3} + 1 + C

Solution steps

\int \frac{12 x ^{2}}{2 x ^{3} + 1} d x

Take the constant out:

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

= 12 \cdot \int \frac{x ^{2}}{2 x ^{3} + 1} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= 12 \cdot \int \frac{1}{6 u} d u

Take the constant out:

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

= 12 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 12 \cdot \frac{1}{6} ln ∣ u ∣

Substitute back

u = 2 x^{3} + 1

= 12 \cdot \frac{1}{6} ln 2 x^{3} + 1

Simplify

12 \cdot \frac{1}{6} ln 2 x^{3} + 1 : 2 ln 2 x^{3} + 1

= 2 ln 2 x^{3} + 1

Add a constant to the solution

= 2 ln 2 x^{3} + 1 + C

\frac{8 x + x ^{2}}{( 4 + x ) ^{2}}

\int \frac{x}{4 - 5 x ^{2}} d x

f (x) = \frac{x}{x + 5}

6 cos (x)

y^{^{''}} + 16 y = - 32 sec (4 t)

What is the integral of (12x^2)/(2x^3+1) ?
The integral of (12x^2)/(2x^3+1) is 2ln|2x^3+1|+C