What is the integral of 1/(x^2sqrt(2-3x)) ?

The integral of 1/(x^2sqrt(2-3x)) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(x^2sqrt(2-3x))

\int \frac{1}{x ^{2} 2 - 3 x} d x

- 48 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )}) + C

Solution steps

\int \frac{1}{x ^{2} 2 - 3 x} d x

Apply u-substitution

: \int - \frac{6}{( u ^{2} - 2 ) ^{2}} d u

= \int - \frac{6}{( u ^{2} - 2 ) ^{2}} d u

Take the constant out:

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

= - 6 \cdot \int \frac{1}{( u ^{2} - 2 ) ^{2}} d u

Apply Integral Substitution

= - 6 \cdot \int \frac{8}{( v ^{2} - 8 ) ^{2}} d v

Take the constant out:

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

= - 6 \cdot 8 \cdot \int \frac{1}{( v ^{2} - 8 ) ^{2}} d v

Take the partial fraction of

\frac{1}{( v ^{2} - 8 ) ^{2}} : \frac{1}{64 2 ( v + 2 2 )} + \frac{1}{32 ( v + 2 2 ) ^{2}} - \frac{1}{64 2 ( v - 2 2 )} + \frac{1}{32 ( v - 2 2 ) ^{2}}

= - 6 \cdot 8 \cdot \int \frac{1}{64 2 ( v + 2 2 )} + \frac{1}{32 ( v + 2 2 ) ^{2}} - \frac{1}{64 2 ( v - 2 2 )} + \frac{1}{32 ( v - 2 2 ) ^{2}} d v

Apply the Sum Rule:

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

= - 6 \cdot 8 (\int \frac{1}{64 2 ( v + 2 2 )} d v + \int \frac{1}{32 ( v + 2 2 ) ^{2}} d v - \int \frac{1}{64 2 ( v - 2 2 )} d v + \int \frac{1}{32 ( v - 2 2 ) ^{2}} d v)

\int \frac{1}{64 2 ( v + 2 2 )} d v = \frac{1}{64 2} ln v + 22

\int \frac{1}{32 ( v + 2 2 ) ^{2}} d v = - \frac{1}{32 ( v + 2 2 )}

\int \frac{1}{64 2 ( v - 2 2 )} d v = \frac{1}{64 2} ln v - 22

\int \frac{1}{32 ( v - 2 2 ) ^{2}} d v = - \frac{1}{32 ( v - 2 2 )}

= - 6 \cdot 8 (\frac{1}{64 2} ln v + 22 - \frac{1}{32 ( v + 2 2 )} - \frac{1}{64 2} ln v - 22 - \frac{1}{32 ( v - 2 2 )})

Substitute back

= - 6 \cdot 8 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )})

Simplify

= - 48 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )})

Add a constant to the solution

= - 48 (\frac{1}{64 2} ln 2 2 - 3 x + 22 - \frac{1}{32 ( 2 2 - 3 x + 2 2 )} - \frac{1}{64 2} ln 2 2 - 3 x - 22 - \frac{1}{32 ( 2 2 - 3 x - 2 2 )}) + C

\int (\frac{1 + ln ( x )}{x}) d x

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

\int x a + b x d x

\int (2 x - 4 x^{3} + 5) d x

\int sec^{2} (\frac{x}{2}) tan^{2} (\frac{x}{2}) d x

What is the integral of 1/(x^2sqrt(2-3x)) ?
The integral of 1/(x^2sqrt(2-3x)) is