What is the integral of (a^{3x}+1)/(a^x+1) ?

The integral of (a^{3x}+1)/(a^x+1) is (a^{2x})/(2ln(a))-(a^x)/(ln(a))+x+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (a^{3x}+1)/(a^x+1)

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

\frac{a ^{2 x}}{2 ln ( a )} - \frac{a ^{x}}{ln ( a )} + x + C

Solution steps

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

Long division

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

= \int a^{2 x} - a^{x} + 1 d x

Apply the Sum Rule:

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

= \int a^{2 x} d x - \int a^{x} d x + \int 1 d x

\int a^{2 x} d x = \frac{a ^{2 x}}{2 ln ( a )}

\int a^{x} d x = \frac{a ^{x}}{ln ( a )}

\int 1 d x = x

= \frac{a ^{2 x}}{2 ln ( a )} - \frac{a ^{x}}{ln ( a )} + x

Add a constant to the solution

= \frac{a ^{2 x}}{2 ln ( a )} - \frac{a ^{x}}{ln ( a )} + x + C

\int (e^{x} sin (y) + yz) d x

\int 6 cos^{3} (4 x) d x

\int \frac{( 3 ^{x} - 2 ^{x} ) ^{2}}{6 ^{x}} d x

\int 6 (x - 1)^{2} d x

What is the integral of (a^{3x}+1)/(a^x+1) ?
The integral of (a^{3x}+1)/(a^x+1) is (a^{2x})/(2ln(a))-(a^x)/(ln(a))+x+C