What is the integral of (x+25)/(x^2+24x+148) ?

The integral of (x+25)/(x^2+24x+148) is 1/2 ln|x^2+24x+148|+13/2 arctan((x+12)/2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x+25)/(x^2+24x+148)

\int \frac{x + 25}{x ^{2} + 24 x + 148} d x

\frac{1}{2} ln x^{2} + 24 x + 148 + \frac{13}{2} arctan (\frac{x + 12}{2}) + C

Solution steps

\int \frac{x + 25}{x ^{2} + 24 x + 148} d x

Apply u-substitution

: \int \frac{u}{u ^{2} - 26 u + 173} d u

= \int \frac{u}{u ^{2} - 26 u + 173} d u

Complete the square

u^{2} - 26 u + 173 : (u - 13)^{2} + 4

= \int \frac{u}{( u - 13 ) ^{2} + 4} d u

Apply u-substitution

: \int \frac{v + 13}{v ^{2} + 4} d v

= \int \frac{v + 13}{v ^{2} + 4} d v

Expand

\frac{v + 13}{v ^{2} + 4} : \frac{v}{v ^{2} + 4} + \frac{13}{v ^{2} + 4}

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{v}{v ^{2} + 4} d v + \int \frac{13}{v ^{2} + 4} d v

\int \frac{v}{v ^{2} + 4} d v = \frac{1}{2} ln v^{2} + 4

\int \frac{13}{v ^{2} + 4} d v = \frac{13}{2} arctan (\frac{v}{2})

= \frac{1}{2} ln v^{2} + 4 + \frac{13}{2} arctan (\frac{v}{2})

Substitute back

= \frac{1}{2} ln (x + 25 - 13)^{2} + 4 + \frac{13}{2} arctan (\frac{x + 25 - 13}{2})

Simplify

\frac{1}{2} ln (x + 25 - 13)^{2} + 4 + \frac{13}{2} arctan (\frac{x + 25 - 13}{2}) : \frac{1}{2} ln x^{2} + 24 x + 148 + \frac{13}{2} arctan (\frac{x + 12}{2})

= \frac{1}{2} ln x^{2} + 24 x + 148 + \frac{13}{2} arctan (\frac{x + 12}{2})

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 24 x + 148 + \frac{13}{2} arctan (\frac{x + 12}{2}) + C

y^{^{'}} = \frac{( cos ( 18 x ) )}{( e ^{(18 \cdot y)} )}

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

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

\int_{- 1}^{0} \frac{4}{1 + ( 2 x + 1 ) ^{2}} d x

x^{3} + 9 x

What is the integral of (x+25)/(x^2+24x+148) ?
The integral of (x+25)/(x^2+24x+148) is 1/2 ln|x^2+24x+148|+13/2 arctan((x+12)/2)+C