What is the integral of 3/(5ysqrt(y^2-1)) ?

The integral of 3/(5ysqrt(y^2-1)) is 3/5 arctan(sqrt(y^2-1))+C

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integral of 3/(5ysqrt(y^2-1))

\int \frac{3}{5 y y ^{2} - 1} d y

\frac{3}{5} arctan (y^{2} - 1) + C

Solution steps

\int \frac{3}{5 y y ^{2} - 1} d y

Take the constant out:

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

= \frac{3}{5} \cdot \int \frac{1}{y y ^{2} - 1} d y

Apply u-substitution

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

= \frac{3}{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{3}{5} arctan (u)

Substitute back

u = y^{2} - 1

= \frac{3}{5} arctan (y^{2} - 1)

Add a constant to the solution

= \frac{3}{5} arctan (y^{2} - 1) + C

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

8 e^{x} + x

x \to - 1 lim (\frac{x}{1 - x ^{4}})

\int_{0}^{1} (1 + r)^{3} d r

s \cdot + 5

What is the integral of 3/(5ysqrt(y^2-1)) ?
The integral of 3/(5ysqrt(y^2-1)) is 3/5 arctan(sqrt(y^2-1))+C