What is the integral of (5x^4)/(1+x^{10)} ?

The integral of (5x^4)/(1+x^{10)} is arctan(x^5)+C

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integral of (5x^4)/(1+x^{10)}

\int \frac{5 x ^{4}}{1 + x ^{10}} d x

arctan (x^{5}) + C

Solution steps

\int \frac{5 x ^{4}}{1 + x ^{10}} 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 ^{10}} d x

Apply u-substitution

: \int \frac{1}{5 ( 1 + u ^{2} )} d u

= 5 \cdot \int \frac{1}{5 ( 1 + u ^{2} )} 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}{1 + u ^{2}} d u

Use the common integral:

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

= 5 \cdot \frac{1}{5} arctan (u)

Substitute back

u = x^{5}

= 5 \cdot \frac{1}{5} arctan (x^{5})

Simplify

5 \cdot \frac{1}{5} arctan (x^{5}) : arctan (x^{5})

= arctan (x^{5})

Add a constant to the solution

= arctan (x^{5}) + C

\int x^{2} - 6 x + 10 d x

y^{^{''}} - 5 y^{^{'}} = (8 t - 7) sin (2 t)

x \to 7 lim (x (x - 2))

\frac{d}{d x} (cos (5 x^{2} + 7 x - 1))

2 x \cdot cos (x^{2})

What is the integral of (5x^4)/(1+x^{10)} ?
The integral of (5x^4)/(1+x^{10)} is arctan(x^5)+C