What is the integral of (20y^3)/(5y^4+1) ?

The integral of (20y^3)/(5y^4+1) is ln|5y^4+1|+C

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integral of (20y^3)/(5y^4+1)

\int \frac{20 y ^{3}}{5 y ^{4} + 1} d y

ln 5 y^{4} + 1 + C

Solution steps

\int \frac{20 y ^{3}}{5 y ^{4} + 1} d y

Take the constant out:

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

= 20 \cdot \int \frac{y ^{3}}{5 y ^{4} + 1} d y

Apply u-substitution

: \int \frac{1}{20 u} d u

= 20 \cdot \int \frac{1}{20 u} d u

Take the constant out:

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

= 20 \cdot \frac{1}{20} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 20 \cdot \frac{1}{20} ln ∣ u ∣

Substitute back

u = 5 y^{4} + 1

= 20 \cdot \frac{1}{20} ln 5 y^{4} + 1

Simplify

20 \cdot \frac{1}{20} ln 5 y^{4} + 1 : ln 5 y^{4} + 1

= ln 5 y^{4} + 1

Add a constant to the solution

= ln 5 y^{4} + 1 + C

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What is the integral of (20y^3)/(5y^4+1) ?
The integral of (20y^3)/(5y^4+1) is ln|5y^4+1|+C