What is the integral of (5x^4)/(1+x^5) ?

The integral of (5x^4)/(1+x^5) is ln|1+x^5|+C

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integral of (5x^4)/(1+x^5)

\int \frac{5 x ^{4}}{1 + x ^{5}} d x

ln 1 + x^{5} + C

Solution steps

\int \frac{5 x ^{4}}{1 + x ^{5}} 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 ^{4}}{1 + x ^{5}} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 1 + x^{5}

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

Simplify

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

= ln 1 + x^{5}

Add a constant to the solution

= ln 1 + x^{5} + C

f (x) = 2 x + 1, at x = 3

g (t) = tan (cos (2 t))

(- 1, - 1), (1, 5)

\frac{d}{d x} ((3 x - 1)^{4} (2 x + 1)^{- 3})

x y^{^{'}} - y = x^{2} + 5 x - 1

What is the integral of (5x^4)/(1+x^5) ?
The integral of (5x^4)/(1+x^5) is ln|1+x^5|+C