What is the integral of 1/(5+9x-2x^2) ?

The integral of 1/(5+9x-2x^2) is 1/11 (ln| 2/11+4/11 x|-ln|-20/11+4/11 x|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(5+9x-2x^2)

\int \frac{1}{5 + 9 x - 2 x ^{2}} d x

\frac{1}{11} (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + C

Solution steps

\int \frac{1}{5 + 9 x - 2 x ^{2}} d x

Complete the square

5 + 9 x - 2 x^{2} : - 2 (x - \frac{9}{4})^{2} + \frac{121}{8}

= \int \frac{1}{- 2 ( x - \frac{9}{4} ) ^{2} + \frac{121}{8}} d x

Apply u-substitution

: \int \frac{8}{- 16 u ^{2} + 121} d u

= \int \frac{8}{- 16 u ^{2} + 121} d u

Take the constant out:

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

= 8 \cdot \int \frac{1}{- 16 u ^{2} + 121} d u

Apply Integral Substitution

= 8 \cdot \int \frac{1}{44 ( - v ^{2} + 1 )} d v

Take the constant out:

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

= 8 \cdot \frac{1}{44} \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}

= 8 \cdot \frac{1}{44} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

= 8 \cdot \frac{1}{44} (\frac{ln \frac{4}{11} ( x - \frac{9}{4} ) + 1}{2} - \frac{ln \frac{4}{11} ( x - \frac{9}{4} ) - 1}{2})

Simplify

8 \cdot \frac{1}{44} (\frac{ln \frac{4}{11} ( x - \frac{9}{4} ) + 1}{2} - \frac{ln \frac{4}{11} ( x - \frac{9}{4} ) - 1}{2}) : \frac{1}{11} (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x)

= \frac{1}{11} (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x)

Add a constant to the solution

= \frac{1}{11} (ln \frac{2}{11} + \frac{4}{11} x - ln - \frac{20}{11} + \frac{4}{11} x) + C

\int x^{3} ln (7 x^{2}) d x

\int 4 e^{- 2 y} d y

\int \frac{5 - 2 y}{y ( y - 1 ) ^{2}} d y

\int \frac{e ^{x}}{( e ^{x} + 1 ) ^{8}} d x

\int \frac{ln ( 2 )}{1 + x ^{2}} d x

What is the integral of 1/(5+9x-2x^2) ?
The integral of 1/(5+9x-2x^2) is 1/11 (ln| 2/11+4/11 x|-ln|-20/11+4/11 x|)+C