What is the integral of 8/(xsqrt(64-(ln(x))^2)) ?

The integral of 8/(xsqrt(64-(ln(x))^2)) is 8arcsin(1/8 ln(x))+C

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integral of 8/(xsqrt(64-(ln(x))^2))

\int \frac{8}{x 64 - ( ln ( x ) ) ^{2}} d x

8 arcsin (\frac{1}{8} ln (x)) + C

Solution steps

\int \frac{8}{x 64 - ( ln ( x ) ) ^{2}} d x

Take the constant out:

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

= 8 \cdot \int \frac{1}{x 64 - ( ln ( x ) ) ^{2}} d x

Simplify

= 8 \cdot \int \frac{1}{x 64 - ln ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{64 - u ^{2}} d u

= 8 \cdot \int \frac{1}{64 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 1 d v

= 8 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 8 \cdot 1 \cdot v

Substitute back

= 8 \cdot 1 \cdot arcsin (\frac{1}{8} ln (x))

Simplify

= 8 arcsin (\frac{1}{8} ln (x))

Add a constant to the solution

= 8 arcsin (\frac{1}{8} ln (x)) + C

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What is the integral of 8/(xsqrt(64-(ln(x))^2)) ?
The integral of 8/(xsqrt(64-(ln(x))^2)) is 8arcsin(1/8 ln(x))+C