What is the integral of (5x^5)/(5+x^6) ?

The integral of (5x^5)/(5+x^6) is 5/6 ln|5+x^6|+C

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integral of (5x^5)/(5+x^6)

\int \frac{5 x ^{5}}{5 + x ^{6}} d x

\frac{5}{6} ln 5 + x^{6} + C

Solution steps

\int \frac{5 x ^{5}}{5 + x ^{6}} d x

Take the constant out:

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

= 5 \cdot \int \frac{x ^{5}}{5 + x ^{6}} d x

Apply u-substitution

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

= 5 \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

= 5 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{6} ln ∣ u ∣

Substitute back

u = 5 + x^{6}

= 5 \cdot \frac{1}{6} ln 5 + x^{6}

Simplify

5 \cdot \frac{1}{6} ln 5 + x^{6} : \frac{5}{6} ln 5 + x^{6}

= \frac{5}{6} ln 5 + x^{6}

Add a constant to the solution

= \frac{5}{6} ln 5 + x^{6} + C

4 x^{2} + 2 x, at - 3

\int \frac{r}{r ^{2} + z ^{2}} d r

\frac{d}{d x} (\frac{1}{1 + \frac{1}{x}})

x \to 3 lim (x^{2} + 5 x)

\frac{d y}{d x} = \frac{1}{11} y cos^{2} (y)

What is the integral of (5x^5)/(5+x^6) ?
The integral of (5x^5)/(5+x^6) is 5/6 ln|5+x^6|+C