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

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

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

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

ln x^{4} + 7 + C

Solution steps

\int \frac{4 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

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

Apply u-substitution

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

= 4 \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 \cdot \frac{1}{4} ln ∣ u ∣

Substitute back

u = x^{4} + 7

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

Simplify

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

= ln x^{4} + 7

Add a constant to the solution

= ln x^{4} + 7 + C

f (x) = 2 x^{2} + 5 x

\int 2 t d t

- x^{2} + 4 x + 1, x + 1

\int \frac{( y - 3 )}{y} d y

\frac{1}{x ^{2}}

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