What is the integral of (3x^2-5x+1)/((x-4)) ?

The integral of (3x^2-5x+1)/((x-4)) is 19x+29ln|x-4|+3/2 (x-4)^2-76+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (3x^2-5x+1)/((x-4))

\int \frac{3 x ^{2} - 5 x + 1}{( x - 4 )} d x

19 x + 29 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 76 + C

Solution steps

\int \frac{3 x ^{2} - 5 x + 1}{x - 4} d x

Expand

\frac{3 x ^{2} - 5 x + 1}{x - 4} : \frac{3 x ^{2}}{x - 4} - \frac{5 x}{x - 4} + \frac{1}{x - 4}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{3 x ^{2}}{x - 4} d x - \int \frac{5 x}{x - 4} d x + \int \frac{1}{x - 4} d x

\int \frac{3 x ^{2}}{x - 4} d x = 3 (\frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣)

\int \frac{5 x}{x - 4} d x = 5 (x - 4 + 4 ln ∣ x - 4 ∣)

\int \frac{1}{x - 4} d x = ln ∣ x - 4 ∣

= 3 (\frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣) - 5 (x - 4 + 4 ln ∣ x - 4 ∣) + ln ∣ x - 4 ∣

Simplify

3 (\frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣) - 5 (x - 4 + 4 ln ∣ x - 4 ∣) + ln ∣ x - 4 ∣ : 19 x + 29 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 76

= 19 x + 29 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 76

Add a constant to the solution

= 19 x + 29 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 76 + C

e^{x^{5}}

f (x) = x^{3} - 3 x^{2}

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

\int \frac{x ^{2} I n ^{3} ( 1 + x ^{3} )}{1 + x ^{3}} d x

\frac{d}{d x} (e^{i x}^{2})

What is the integral of (3x^2-5x+1)/((x-4)) ?
The integral of (3x^2-5x+1)/((x-4)) is 19x+29ln|x-4|+3/2 (x-4)^2-76+C