What is the integral of 1/(3xsqrt(9-x^2)) ?

The integral of 1/(3xsqrt(9-x^2)) is -1/18 (ln|(sqrt(9-x^2))/3+1|-ln|(sqrt(9-x^2))/3-1|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(3xsqrt(9-x^2))

\int \frac{1}{3 x 9 - x ^{2}} d x

- \frac{1}{18} (ln \frac{9 - x ^{2}}{3} + 1 - ln \frac{9 - x ^{2}}{3} - 1) + C

Solution steps

\int \frac{1}{3 x 9 - x ^{2}} d x

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{3} (- \int \frac{1}{- u ^{2} + 9} d u)

Apply Integral Substitution

= \frac{1}{3} (- \int \frac{1}{3 ( - v ^{2} + 1 )} d v)

Take the constant out:

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

= \frac{1}{3} (- \frac{1}{3} \cdot \int \frac{1}{- v ^{2} + 1} d v)

Use the common integral:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= \frac{1}{3} (- \frac{1}{3} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= \frac{1}{3} - \frac{1}{3} \frac{ln \frac{9 - x ^{2}}{3} + 1}{2} - \frac{ln \frac{9 - x ^{2}}{3} - 1}{2}

Simplify

\frac{1}{3} - \frac{1}{3} \frac{ln \frac{9 - x ^{2}}{3} + 1}{2} - \frac{ln \frac{9 - x ^{2}}{3} - 1}{2} : - \frac{1}{18} (ln \frac{9 - x ^{2}}{3} + 1 - ln \frac{9 - x ^{2}}{3} - 1)

= - \frac{1}{18} (ln \frac{9 - x ^{2}}{3} + 1 - ln \frac{9 - x ^{2}}{3} - 1)

Add a constant to the solution

= - \frac{1}{18} (ln \frac{9 - x ^{2}}{3} + 1 - ln \frac{9 - x ^{2}}{3} - 1) + C

\int \frac{( 4 x - 11 )}{x ^{3} - 9 x ^{2}} d x

\int cos (\frac{1}{x}) cos (\frac{3}{x}) \frac{1}{x ^{2}} d x

\int \frac{cot ^{2} ( ln ( x ) )}{x csc ( ln ( x ) )} d x

\int cos^{2} (x) sin^{3} (2 x) d x

\int \frac{1}{x ^{2} 8 x + 1} d x

What is the integral of 1/(3xsqrt(9-x^2)) ?
The integral of 1/(3xsqrt(9-x^2)) is -1/18 (ln|(sqrt(9-x^2))/3+1|-ln|(sqrt(9-x^2))/3-1|)+C