What is the integral of (12x^{11})/(x^{12)+1} ?

The integral of (12x^{11})/(x^{12)+1} is ln|x^{12}+1|+C

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integral of (12x^{11})/(x^{12)+1}

\int \frac{12 x ^{11}}{x ^{12} + 1} d x

ln x^{12} + 1 + C

Solution steps

\int \frac{12 x ^{11}}{x ^{12} + 1} d x

Take the constant out:

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

= 12 \cdot \int \frac{x ^{11}}{x ^{12} + 1} d x

Apply u-substitution

= 12 \cdot \int \frac{1}{12 u} d u

Take the constant out:

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

= 12 \cdot \frac{1}{12} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 12 \cdot \frac{1}{12} ln ∣ u ∣

Substitute back

u = x^{12} + 1

= 12 \cdot \frac{1}{12} ln x^{12} + 1

Simplify

12 \cdot \frac{1}{12} ln x^{12} + 1 : ln x^{12} + 1

= ln x^{12} + 1

Add a constant to the solution

= ln x^{12} + 1 + C

What is the integral of (12x^{11})/(x^{12)+1} ?
The integral of (12x^{11})/(x^{12)+1} is ln|x^{12}+1|+C