What is the integral of ((x^7))/(1+x^{16)} ?

The integral of ((x^7))/(1+x^{16)} is 1/8 arctan(x^8)+C

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integral of ((x^7))/(1+x^{16)}

\int \frac{( x ^{7} )}{1 + x ^{16}} d x

\frac{1}{8} arctan (x^{8}) + C

Solution steps

\int \frac{x ^{7}}{1 + x ^{16}} d x

Apply u-substitution

: \int \frac{1}{8 ( 1 + u ^{2} )} d u

= \int \frac{1}{8 ( 1 + u ^{2} )} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= \frac{1}{8} arctan (u)

Substitute back

u = x^{8}

= \frac{1}{8} arctan (x^{8})

Add a constant to the solution

= \frac{1}{8} arctan (x^{8}) + C

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2 4 - x

\frac{d}{d x} (ka ln (\frac{x + d}{x - d}))

\frac{1}{2} cos (7 x)

\frac{d y}{d x} = y^{2} + 2 y

What is the integral of ((x^7))/(1+x^{16)} ?
The integral of ((x^7))/(1+x^{16)} is 1/8 arctan(x^8)+C