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

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

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

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

2 arcsin (\frac{1}{2} ln (x)) + C

Solution steps

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

Simplify

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

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int 1 d v

= 2 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot v

Substitute back

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

Simplify

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

Add a constant to the solution

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

y^{^{''}} - 8 y^{^{'}} + 12 y = 0

\int_{1}^{2} x y^{3} e^{x^{2} y^{2}} d x

x y^{^{'}} - 2 y = 9 x^{2}

y = 3 x^{3} + 5 x + 4, (2, 38)

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

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