What is the integral of 1/(-y^2+y+20) ?

The integral of 1/(-y^2+y+20) is 1/9 (ln| 8/9+2/9 y|-ln|-10/9+2/9 y|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(-y^2+y+20)

\int \frac{1}{- y ^{2} + y + 20} d y

\frac{1}{9} (ln \frac{8}{9} + \frac{2}{9} y - ln - \frac{10}{9} + \frac{2}{9} y) + C

Solution steps

\int \frac{1}{- y ^{2} + y + 20} d y

Complete the square

- y^{2} + y + 20 : - (y - \frac{1}{2})^{2} + \frac{81}{4}

= \int \frac{1}{- ( y - \frac{1}{2} ) ^{2} + \frac{81}{4}} d y

Apply u-substitution

: \int \frac{4}{- 4 u ^{2} + 81} d u

= \int \frac{4}{- 4 u ^{2} + 81} d u

Take the constant out:

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

= 4 \cdot \int \frac{1}{- 4 u ^{2} + 81} d u

Apply Integral Substitution

= 4 \cdot \int \frac{1}{18 ( - v ^{2} + 1 )} d v

Take the constant out:

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

= 4 \cdot \frac{1}{18} \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}

= 4 \cdot \frac{1}{18} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

= 4 \cdot \frac{1}{18} (\frac{ln \frac{2}{9} ( y - \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{9} ( y - \frac{1}{2} ) - 1}{2})

Simplify

4 \cdot \frac{1}{18} (\frac{ln \frac{2}{9} ( y - \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{9} ( y - \frac{1}{2} ) - 1}{2}) : \frac{1}{9} (ln \frac{8}{9} + \frac{2}{9} y - ln - \frac{10}{9} + \frac{2}{9} y)

= \frac{1}{9} (ln \frac{8}{9} + \frac{2}{9} y - ln - \frac{10}{9} + \frac{2}{9} y)

Add a constant to the solution

= \frac{1}{9} (ln \frac{8}{9} + \frac{2}{9} y - ln - \frac{10}{9} + \frac{2}{9} y) + C

x \to 2 lim (\frac{- 1}{0})

y = (1 + 5 x)^{10}, (0, 1)

\frac{d}{d x} (- \frac{2 x}{( 3 + x ^{2} ) ^{2}})

\int (\frac{2 x + y + z}{( x + y ) ( x + z )}) d x

x^{4} cos (x^{3})

What is the integral of 1/(-y^2+y+20) ?
The integral of 1/(-y^2+y+20) is 1/9 (ln| 8/9+2/9 y|-ln|-10/9+2/9 y|)+C