What is the integral of (3x^2)/(x^3+7) ?

The integral of (3x^2)/(x^3+7) is ln|x^3+7|+C

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integral of (3x^2)/(x^3+7)

\int \frac{3 x ^{2}}{x ^{3} + 7} d x

ln x^{3} + 7 + C

Solution steps

\int \frac{3 x ^{2}}{x ^{3} + 7} 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} + 7} 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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = x^{3} + 7

= 3 \cdot \frac{1}{3} ln x^{3} + 7

Simplify

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

= ln x^{3} + 7

Add a constant to the solution

= ln x^{3} + 7 + C

\frac{\partial}{\partial x} (ln (cos (x)))

\frac{( x ^{2} )}{x ^{2} + 27}

\frac{d}{d x} (\frac{cos ( x )}{lo g _{3} ( x )})

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

\frac{\partial}{\partial x} (\frac{r x ( K ( r , x ) - x )}{K ( r , x ) + a x})

What is the integral of (3x^2)/(x^3+7) ?
The integral of (3x^2)/(x^3+7) is ln|x^3+7|+C