What is the integral of 5/(xsqrt(25-(ln(x))^2)) ?

The integral of 5/(xsqrt(25-(ln(x))^2)) is 5arcsin(1/5 ln(x))+C

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integral of 5/(xsqrt(25-(ln(x))^2))

\int \frac{5}{x 25 - ( ln ( x ) ) ^{2}} d x

5 arcsin (\frac{1}{5} ln (x)) + C

Solution steps

\int \frac{5}{x 25 - ( ln ( x ) ) ^{2}} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x 25 - ( ln ( x ) ) ^{2}} d x

Simplify

= 5 \cdot \int \frac{1}{x 25 - ln ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{25 - u ^{2}} d u

= 5 \cdot \int \frac{1}{25 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 1 d v

= 5 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 5 \cdot 1 \cdot v

Substitute back

= 5 \cdot 1 \cdot arcsin (\frac{1}{5} ln (x))

Simplify

= 5 arcsin (\frac{1}{5} ln (x))

Add a constant to the solution

= 5 arcsin (\frac{1}{5} ln (x)) + C

\int cos (x) e^{- 3 x} d x

y = \frac{x}{25 + x ^{2}}, (0, 0)

\int (4 e^{2 t} i + 3 e^{t} j) d t

\frac{\partial}{\partial x} (\frac{x}{x ^{2} + 2 y ^{2}} + 3 x)

x \to 1 lim (2 ∣ x - 1 ∣ - 1)

What is the integral of 5/(xsqrt(25-(ln(x))^2)) ?
The integral of 5/(xsqrt(25-(ln(x))^2)) is 5arcsin(1/5 ln(x))+C