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

The integral of 3/(xsqrt(9-(ln(x))^2)) is 3arcsin(1/3 ln(x))+C

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

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

3 arcsin (\frac{1}{3} ln (x)) + C

Solution steps

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

Simplify

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

Apply u-substitution

: \int \frac{1}{9 - u ^{2}} d u

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

Apply Trigonometric Substitution

: \int 1 d v

= 3 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 3 \cdot 1 \cdot v

Substitute back

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

Simplify

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

Add a constant to the solution

= 3 arcsin (\frac{1}{3} ln (x)) + C

f (x) = x^{2} - 10 x + 2, at x = 0

x \to 0 lim (csc (x) - cot (x))

\int 0.1 e^{- 0.1 x} d x

\frac{d y}{d t} = sin (3 t) - 2 y

x \to 9 + lim (\frac{5}{x - 9})

What is the integral of 3/(xsqrt(9-(ln(x))^2)) ?
The integral of 3/(xsqrt(9-(ln(x))^2)) is 3arcsin(1/3 ln(x))+C