What is the integral of (x^{15})/(1+x^{16)} ?

The integral of (x^{15})/(1+x^{16)} is 1/16 ln|1+x^{16}|+C

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integral of (x^{15})/(1+x^{16)}

\int \frac{x ^{15}}{1 + x ^{16}} d x

\frac{1}{16} ln 1 + x^{16} + C

Solution steps

\int \frac{x ^{15}}{1 + x ^{16}} d x

Apply u-substitution

: \int \frac{1}{16 u} d u

= \int \frac{1}{16 u} d u

Take the constant out:

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

= \frac{1}{16} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{16} ln ∣ u ∣

Substitute back

u = 1 + x^{16}

= \frac{1}{16} ln 1 + x^{16}

Add a constant to the solution

= \frac{1}{16} ln 1 + x^{16} + C

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\frac{d}{d t} (a cos (3 t))

What is the integral of (x^{15})/(1+x^{16)} ?
The integral of (x^{15})/(1+x^{16)} is 1/16 ln|1+x^{16}|+C