What is the integral of 3/(xsqrt(4-9ln^2(x))) ?

The integral of 3/(xsqrt(4-9ln^2(x))) is arcsin(3/2 ln(x))+C

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integral of 3/(xsqrt(4-9ln^2(x)))

\int \frac{3}{x 4 - 9 ln ^{2} ( x )} d x

arcsin (\frac{3}{2} ln (x)) + C

Solution steps

\int \frac{3}{x 4 - 9 ln ^{2} ( x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{x 4 - 9 ln ^{2} ( x )} d x

Apply u-substitution

= 3 \cdot \int \frac{1}{4 - 9 u ^{2}} d u

Apply Trigonometric Substitution

= 3 \cdot \int \frac{1}{3} d v

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 3 \cdot \frac{1}{3} \cdot 1 \cdot v

Substitute back

= 3 \cdot \frac{1}{3} \cdot 1 \cdot arcsin (\frac{3}{2} ln (x))

Simplify

3 \cdot \frac{1}{3} \cdot 1 \cdot arcsin (\frac{3}{2} ln (x)) : arcsin (\frac{3}{2} ln (x))

= arcsin (\frac{3}{2} ln (x))

Add a constant to the solution

= arcsin (\frac{3}{2} ln (x)) + C

d er i v a t i v e \frac{4 x ^{\frac{3}{2}}}{x}

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\frac{\partial}{\partial x} (5 x^{7} cos (y^{8}))

x \to 0 lim (1 + sin (x) - e^{x})

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

What is the integral of 3/(xsqrt(4-9ln^2(x))) ?
The integral of 3/(xsqrt(4-9ln^2(x))) is arcsin(3/2 ln(x))+C