What is the integral of (2^{sqrt(x-3)})/(sqrt(x-3)) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2^{sqrt(x-3)})/(sqrt(x-3))

\int \frac{2 ^{x - 3}}{x - 3} d x

\frac{2 \cdot 2 ^{x - 3}}{ln ( 2 )} + C

Solution steps

\int \frac{2 ^{x - 3}}{x - 3} d x

Apply u-substitution

: \int \frac{2 ^{u}}{u} d u

= \int \frac{2 ^{u}}{u} d u

Apply u-substitution

: \int \frac{v}{ln ( 2 ) \cdot 2 ^{\frac{l n ( v ) - l n ( 2 )}{l n ( 2 )}}} d v

= \int \frac{v}{ln ( 2 ) \cdot 2 ^{\frac{l n ( v ) - l n ( 2 )}{l n ( 2 )}}} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{ln ( 2 )} \cdot \int \frac{v}{2 ^{\frac{l n ( v ) - l n ( 2 )}{l n ( 2 )}}} d v

Apply Integration By Parts

: \frac{v ^{2}}{2 ^{\frac{l n ( v )}{l n ( 2 )}}} - \int - 1 d v

= \frac{1}{ln ( 2 )} (\frac{v ^{2}}{2 ^{\frac{l n ( v )}{l n ( 2 )}}} - \int - 1 d v)

\int - 1 d v = - v

= \frac{1}{ln ( 2 )} (\frac{v ^{2}}{2 ^{\frac{l n ( v )}{l n ( 2 )}}} - (- v))

Substitute back

= \frac{1}{ln ( 2 )} \frac{( 2 ^{x - 3} ) ^{2}}{2 ^{\frac{l n ( 2 ^{x - 3} )}{l n ( 2 )}}} - (- 2^{x - 3})

Simplify

\frac{1}{ln ( 2 )} \frac{( 2 ^{x - 3} ) ^{2}}{2 ^{\frac{l n ( 2 ^{x - 3} )}{l n ( 2 )}}} - (- 2^{x - 3}) : \frac{2 \cdot 2 ^{x - 3}}{ln ( 2 )}

= \frac{2 \cdot 2 ^{x - 3}}{ln ( 2 )}

Add a constant to the solution

= \frac{2 \cdot 2 ^{x - 3}}{ln ( 2 )} + C

\int \frac{x}{x + 8} d x

\int x 5 x + 4 d x

\int e^{7 x} sin (e^{7 x}) d x

\int \frac{1}{( sin ( x ) + cos ( x ) )} d x

\int \frac{10}{( 1 + x ) ^{2}} d x

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