What is the integral of 9/(xsqrt(81-(ln(x))^2)) ?

The integral of 9/(xsqrt(81-(ln(x))^2)) is 9arcsin(1/9 ln(x))+C

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integral of 9/(xsqrt(81-(ln(x))^2))

\int \frac{9}{x 81 - ( ln ( x ) ) ^{2}} d x

9 arcsin (\frac{1}{9} ln (x)) + C

Solution steps

\int \frac{9}{x 81 - ( ln ( x ) ) ^{2}} d x

Take the constant out:

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

= 9 \cdot \int \frac{1}{x 81 - ( ln ( x ) ) ^{2}} d x

Simplify

= 9 \cdot \int \frac{1}{x 81 - ln ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{81 - u ^{2}} d u

= 9 \cdot \int \frac{1}{81 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 1 d v

= 9 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 9 \cdot 1 \cdot v

Substitute back

= 9 \cdot 1 \cdot arcsin (\frac{1}{9} ln (x))

Simplify

= 9 arcsin (\frac{1}{9} ln (x))

Add a constant to the solution

= 9 arcsin (\frac{1}{9} ln (x)) + C

\frac{d}{d x} (x + x)

3 x - 6 y x^{2} + 1 (\frac{d y}{d x}) = 0, y (0) = 3

x^{\frac{12}{7}}

(\frac{x}{7} + 1)^{\frac{3}{2}}

\int_{1}^{3} x^{2} + \frac{1}{4 x ^{2}} d x

What is the integral of 9/(xsqrt(81-(ln(x))^2)) ?
The integral of 9/(xsqrt(81-(ln(x))^2)) is 9arcsin(1/9 ln(x))+C