en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Calculus >

integral of ln(1+x^4)

Solution

\int ln (1 + x^{4}) d x

Solution

x ln (1 + x^{4}) + \frac{1}{2} ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1) - \frac{1}{2} ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1) - 4 x + C

Solution steps

\int ln (1 + x^{4}) d x

Apply Integration By Parts

= x ln (1 + x^{4}) - \int \frac{4 x ^{4}}{1 + x ^{4}} d x

\int \frac{4 x ^{4}}{1 + x ^{4}} d x = - \frac{1}{2} ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1) + \frac{1}{2} ln 2 x^{2} - 22 x + 2 - 2 arctan (2 x - 1) + 4 x

= x ln (1 + x^{4}) - (- \frac{1}{2} ln 2 x^{2} + 22 x + 2 - 2 arctan (2 x + 1) + \frac{1}{2} ln 2 x^{2} - 22 x + 2 - 2 arctan (2 x - 1) + 4 x)

Simplify

= x ln (1 + x^{4}) + \frac{1}{2} ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1) - \frac{1}{2} ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1) - 4 x

Add a constant to the solution

= x ln (1 + x^{4}) + \frac{1}{2} ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1) - \frac{1}{2} ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1) - 4 x + C

Graph

Popular Examples

limit as n approaches infinity of-1

integral from 1 to 2 of-1/(16z)+1/(4z)

derivative of 20x(2x^2+7)^4

derivative of-5x^2+6x

integral from 0 to pi of 5sin^2(2x)

Frequently Asked Questions (FAQ)

What is the integral of ln(1+x^4) ?
The integral of ln(1+x^4) is