What is the integral of (y^3-4y+3)/(y+5) ?

The integral of (y^3-4y+3)/(y+5) is y^3+15y^2+146y+1/3 (y+5)^3-102ln|y+5|-(y+5)^3-15/2 (y+5)^2+378+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (y^3-4y+3)/(y+5)

\int \frac{y ^{3} - 4 y + 3}{y + 5} d y

y^{3} + 15 y^{2} + 146 y + \frac{1}{3} (y + 5)^{3} - 102 ln ∣ y + 5 ∣ - (y + 5)^{3} - \frac{15}{2} (y + 5)^{2} + 378 + C

Solution steps

\int \frac{y ^{3} - 4 y + 3}{y + 5} d y

Apply u-substitution

: \int \frac{- 4 u + ( u - 5 ) ^{3} + 23}{u} d u

= \int \frac{- 4 u + ( u - 5 ) ^{3} + 23}{u} d u

Apply Integration By Parts

: - 4 u + (u - 5)^{3} + 23 - \int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u

= - 4 u + (u - 5)^{3} + 23 - \int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u

\int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u = u^{3} - \frac{u ^{3}}{3} - 15 u^{2} + \frac{15 u ^{2}}{2} + 102 ln ∣ u ∣

= - 4 u + (u - 5)^{3} + 23 - (u^{3} - \frac{u ^{3}}{3} - 15 u^{2} + \frac{15 u ^{2}}{2} + 102 ln ∣ u ∣)

Substitute back

u = y + 5

= - 4 (y + 5) + (y + 5 - 5)^{3} + 23 - ((y + 5)^{3} - \frac{( y + 5 ) ^{3}}{3} - 15 (y + 5)^{2} + \frac{15 ( y + 5 ) ^{2}}{2} + 102 ln ∣ y + 5 ∣)

Simplify

- 4 (y + 5) + (y + 5 - 5)^{3} + 23 - ((y + 5)^{3} - \frac{( y + 5 ) ^{3}}{3} - 15 (y + 5)^{2} + \frac{15 ( y + 5 ) ^{2}}{2} + 102 ln ∣ y + 5 ∣) : y^{3} + 15 y^{2} + 146 y + \frac{1}{3} (y + 5)^{3} - 102 ln ∣ y + 5 ∣ - (y + 5)^{3} - \frac{15}{2} (y + 5)^{2} + 378

= y^{3} + 15 y^{2} + 146 y + \frac{1}{3} (y + 5)^{3} - 102 ln ∣ y + 5 ∣ - (y + 5)^{3} - \frac{15}{2} (y + 5)^{2} + 378

Add a constant to the solution

= y^{3} + 15 y^{2} + 146 y + \frac{1}{3} (y + 5)^{3} - 102 ln ∣ y + 5 ∣ - (y + 5)^{3} - \frac{15}{2} (y + 5)^{2} + 378 + C

\int \frac{1}{x + 2 x ^{3}} d x

\int \frac{1}{2 x ^{2} x ^{2} - 16} d x

\int \frac{7}{w ^{2} + 3 w + 3} d w

\int 4 cot^{3} (x) csc (x) d x

\int 1 + 3 ln (x + 1) d x

What is the integral of (y^3-4y+3)/(y+5) ?
The integral of (y^3-4y+3)/(y+5) is y^3+15y^2+146y+1/3 (y+5)^3-102ln|y+5|-(y+5)^3-15/2 (y+5)^2+378+C