What is the integral of (12x^3)/(x^4+7) ?

The integral of (12x^3)/(x^4+7) is 3ln|x^4+7|+C

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integral of (12x^3)/(x^4+7)

\int \frac{12 x ^{3}}{x ^{4} + 7} d x

3 ln x^{4} + 7 + C

Solution steps

\int \frac{12 x ^{3}}{x ^{4} + 7} 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 ^{3}}{x ^{4} + 7} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= 12 \cdot \int \frac{1}{4 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}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 12 \cdot \frac{1}{4} ln ∣ u ∣

Substitute back

u = x^{4} + 7

= 12 \cdot \frac{1}{4} ln x^{4} + 7

Simplify

12 \cdot \frac{1}{4} ln x^{4} + 7 : 3 ln x^{4} + 7

= 3 ln x^{4} + 7

Add a constant to the solution

= 3 ln x^{4} + 7 + C

y = x^{3}, at x = - 1

f (x) = \frac{2 x}{sin ( x ) + cos ( x )}

\int \frac{( y ^{2} + 1 ) ^{2}}{y ^{2}} d y

x \to 2 lim (\frac{3}{x})

\frac{\partial}{\partial x} (3 x e^{y} - x^{3} - e^{3 y})

What is the integral of (12x^3)/(x^4+7) ?
The integral of (12x^3)/(x^4+7) is 3ln|x^4+7|+C