en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (t^5-1)/(t^4-1)

Solution

\int \frac{t ^{5} - 1}{t ^{4} - 1} d t

Solution

\frac{t ^{2}}{2} - \frac{1}{4} ln t^{2} + 1 + \frac{1}{4} ln t^{2} - 1 + \frac{1}{2} arctan (t) + \frac{1}{4} ln ∣ t + 1 ∣ - \frac{1}{4} ln ∣ t - 1 ∣ + C

Solution steps

\int \frac{t ^{5} - 1}{t ^{4} - 1} d t

Expand

\frac{t ^{5} - 1}{t ^{4} - 1} : \frac{t ^{5}}{t ^{4} - 1} - \frac{1}{t ^{4} - 1}

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 ^{5}}{t ^{4} - 1} d t - \int \frac{1}{t ^{4} - 1} d t

\int \frac{t ^{5}}{t ^{4} - 1} d t = \frac{t ^{2}}{2} - \frac{1}{4} (ln t^{2} + 1 - ln t^{2} - 1)

\int \frac{1}{t ^{4} - 1} d t = - \frac{1}{2} arctan (t) - \frac{1}{4} ln ∣ t + 1 ∣ + \frac{1}{4} ln ∣ t - 1 ∣

= \frac{t ^{2}}{2} - \frac{1}{4} (ln t^{2} + 1 - ln t^{2} - 1) - (- \frac{1}{2} arctan (t) - \frac{1}{4} ln ∣ t + 1 ∣ + \frac{1}{4} ln ∣ t - 1 ∣)

Simplify

\frac{t ^{2}}{2} - \frac{1}{4} (ln t^{2} + 1 - ln t^{2} - 1) - (- \frac{1}{2} arctan (t) - \frac{1}{4} ln ∣ t + 1 ∣ + \frac{1}{4} ln ∣ t - 1 ∣) : \frac{t ^{2}}{2} - \frac{1}{4} ln t^{2} + 1 + \frac{1}{4} ln t^{2} - 1 + \frac{1}{2} arctan (t) + \frac{1}{4} ln ∣ t + 1 ∣ - \frac{1}{4} ln ∣ t - 1 ∣

= \frac{t ^{2}}{2} - \frac{1}{4} ln t^{2} + 1 + \frac{1}{4} ln t^{2} - 1 + \frac{1}{2} arctan (t) + \frac{1}{4} ln ∣ t + 1 ∣ - \frac{1}{4} ln ∣ t - 1 ∣

Add a constant to the solution

= \frac{t ^{2}}{2} - \frac{1}{4} ln t^{2} + 1 + \frac{1}{4} ln t^{2} - 1 + \frac{1}{2} arctan (t) + \frac{1}{4} ln ∣ t + 1 ∣ - \frac{1}{4} ln ∣ t - 1 ∣ + C

Popular Examples

integral of x/(sqrt(9+8x^2-x^4))

\int \frac{x}{9 + 8 x ^{2} - x ^{4}} d x

integral of x-sin(x)

\int x - sin (x) d x

integral of z/(e^{9z)}

\int \frac{z}{e ^{9 z}} d z

integral of x^5ln(x^7)

\int x^{5} ln (x^{7}) d x

integral of (e^{sqrt(2x)})/(sqrt(2x))

\int \frac{e ^{2 x}}{2 x} d x

Frequently Asked Questions (FAQ)

What is the integral of (t^5-1)/(t^4-1) ?
The integral of (t^5-1)/(t^4-1) is