What is the integral of 2/(xsqrt(1-4ln^2(x))) ?

The integral of 2/(xsqrt(1-4ln^2(x))) is arcsin(2ln(x))+C

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integral of 2/(xsqrt(1-4ln^2(x)))

\int \frac{2}{x 1 - 4 ln ^{2} ( x )} d x

arcsin (2 ln (x)) + C

Solution steps

\int \frac{2}{x 1 - 4 ln ^{2} ( x )} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{x 1 - 4 ln ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{1 - 4 u ^{2}} d u

= 2 \cdot \int \frac{1}{1 - 4 u ^{2}} d u

Apply Trigonometric Substitution

: \int \frac{1}{2} d v

= 2 \cdot \int \frac{1}{2} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot \frac{1}{2} \cdot 1 \cdot v

Substitute back

= 2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (2 ln (x))

Simplify

2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (2 ln (x)) : arcsin (2 ln (x))

= arcsin (2 ln (x))

Add a constant to the solution

= arcsin (2 ln (x)) + C

ln (e^{r x} + s)

f (x) = x^{2} - 2 x - 2

f (x) = e^{x} - e^{- x}

\frac{\partial}{\partial x} (1 0^{- 2 x})

y^{^{'}} = \frac{x ^{3} y ^{3} + 2 x ^{3}}{y ^{2}}

What is the integral of 2/(xsqrt(1-4ln^2(x))) ?
The integral of 2/(xsqrt(1-4ln^2(x))) is arcsin(2ln(x))+C