What is the integral of (t^2+8)/(t^2-5t+6) ?

The integral of (t^2+8)/(t^2-5t+6) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (t^2+8)/(t^2-5t+6)

\int \frac{t ^{2} + 8}{t ^{2} - 5 t + 6} d t

4 (\frac{5}{8} ln 4 t^{2} - 20 t + 24 - \frac{13}{8} ln ∣ 2 t - 4 ∣ + \frac{13}{8} ln ∣ 2 t - 6 ∣) + t + 8 (- ln ∣ t - 2 ∣ + ln ∣ t - 3 ∣) + C

Solution steps

\int \frac{t ^{2} + 8}{t ^{2} - 5 t + 6} d t

Expand

\frac{t ^{2} + 8}{t ^{2} - 5 t + 6} : \frac{t ^{2}}{t ^{2} - 5 t + 6} + \frac{8}{t ^{2} - 5 t + 6}

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{t ^{2}}{t ^{2} - 5 t + 6} d t + \int \frac{8}{t ^{2} - 5 t + 6} d t

\int \frac{t ^{2}}{t ^{2} - 5 t + 6} d t = 4 (\frac{5}{8} ln 4 t^{2} - 20 t + 24 - \frac{13}{8} ln ∣ 2 t - 4 ∣ + \frac{13}{8} ln ∣ 2 t - 6 ∣) + t

\int \frac{8}{t ^{2} - 5 t + 6} d t = 8 (- ln ∣ t - 2 ∣ + ln ∣ t - 3 ∣)

= 4 (\frac{5}{8} ln 4 t^{2} - 20 t + 24 - \frac{13}{8} ln ∣ 2 t - 4 ∣ + \frac{13}{8} ln ∣ 2 t - 6 ∣) + t + 8 (- ln ∣ t - 2 ∣ + ln ∣ t - 3 ∣)

Add a constant to the solution

= 4 (\frac{5}{8} ln 4 t^{2} - 20 t + 24 - \frac{13}{8} ln ∣ 2 t - 4 ∣ + \frac{13}{8} ln ∣ 2 t - 6 ∣) + t + 8 (- ln ∣ t - 2 ∣ + ln ∣ t - 3 ∣) + C

3 x^{2} + 4 x

\int ∣ x - 5 ∣ d x

x \to 0 lim (\frac{1}{ln ∣ x ∣})

\int \frac{15 x e ^{2 x}}{( 1 + 2 x ) ^{2}} d x

\frac{d y}{d x}, e^{5 y^{2}} = x^{3} + 3

What is the integral of (t^2+8)/(t^2-5t+6) ?
The integral of (t^2+8)/(t^2-5t+6) is