What is the integral of 1/(5x^2+1) ?

The integral of 1/(5x^2+1) is 1/(sqrt(5))arctan(sqrt(5)x)+C

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integral of 1/(5x^2+1)

\int \frac{1}{5 x ^{2} + 1} d x

\frac{1}{5} arctan (5 x) + C

Solution steps

\int \frac{1}{5 x ^{2} + 1} d x

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 5 x

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

Add a constant to the solution

= \frac{1}{5} arctan (5 x) + C

x \to + 100 lim (7)

\int \frac{4 + x - 3 x}{x} d x

f (x) = - 2 x^{4} + 5 x^{2} - 3, (1, 0)

\frac{d ^{2}}{d x ^{2}} (\frac{1}{x})

\frac{\partial}{\partial x} (\frac{y}{2 x + y})

What is the integral of 1/(5x^2+1) ?
The integral of 1/(5x^2+1) is 1/(sqrt(5))arctan(sqrt(5)x)+C