What is the integral of ((4x+13))/(x^2+3x-10) ?

The integral of ((4x+13))/(x^2+3x-10) is 2ln|4x^2+12x-40|-ln| 10/7+2/7 x|+ln|-4/7+2/7 x|+C

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integral of ((4x+13))/(x^2+3x-10)

\int \frac{( 4 x + 13 )}{x ^{2} + 3 x - 10} d x

2 ln 4 x^{2} + 12 x - 40 - ln \frac{10}{7} + \frac{2}{7} x + ln - \frac{4}{7} + \frac{2}{7} x + C

Solution steps

\int \frac{4 x + 13}{x ^{2} + 3 x - 10} d x

Complete the square

x^{2} + 3 x - 10 : (x + \frac{3}{2})^{2} - \frac{49}{4}

= \int \frac{4 x + 13}{( x + \frac{3}{2} ) ^{2} - \frac{49}{4}} d x

Apply u-substitution

: \int \frac{4 ( 4 u + 7 )}{4 u ^{2} - 49} d u

= \int \frac{4 ( 4 u + 7 )}{4 u ^{2} - 49} d u

Take the constant out:

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

= 4 \cdot \int \frac{4 u + 7}{4 u ^{2} - 49} d u

Expand

\frac{4 u + 7}{4 u ^{2} - 49} : \frac{4 u}{4 u ^{2} - 49} + \frac{7}{4 u ^{2} - 49}

Apply the Sum Rule:

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

= 4 (\int \frac{4 u}{4 u ^{2} - 49} d u + \int \frac{7}{4 u ^{2} - 49} d u)

\int \frac{4 u}{4 u ^{2} - 49} d u = \frac{1}{2} ln 4 u^{2} - 49

\int \frac{7}{4 u ^{2} - 49} d u = - \frac{1}{4} (ln \frac{2}{7} u + 1 - ln \frac{2}{7} u - 1)

= 4 (\frac{1}{2} ln 4 u^{2} - 49 - \frac{1}{4} (ln \frac{2}{7} u + 1 - ln \frac{2}{7} u - 1))

Substitute back

u = x + \frac{3}{2}

= 4 (\frac{1}{2} ln 4 (x + \frac{3}{2})^{2} - 49 - \frac{1}{4} (ln \frac{2}{7} (x + \frac{3}{2}) + 1 - ln \frac{2}{7} (x + \frac{3}{2}) - 1))

Simplify

4 (\frac{1}{2} ln 4 (x + \frac{3}{2})^{2} - 49 - \frac{1}{4} (ln \frac{2}{7} (x + \frac{3}{2}) + 1 - ln \frac{2}{7} (x + \frac{3}{2}) - 1)) : 2 ln 4 x^{2} + 12 x - 40 - ln \frac{10}{7} + \frac{2}{7} x + ln - \frac{4}{7} + \frac{2}{7} x

= 2 ln 4 x^{2} + 12 x - 40 - ln \frac{10}{7} + \frac{2}{7} x + ln - \frac{4}{7} + \frac{2}{7} x

Add a constant to the solution

= 2 ln 4 x^{2} + 12 x - 40 - ln \frac{10}{7} + \frac{2}{7} x + ln - \frac{4}{7} + \frac{2}{7} x + C

\int_{0}^{\frac{π}{2}} \frac{5 cos ( x )}{sin ( x )} d x

\frac{d y}{d t} = \frac{4 t ^{8}}{e + t ^{9}}

\frac{d}{d x} (\frac{- 2}{( x - 1 ) ^{2}})

y^{^{'}} = e^{5 x} - 9 x

\frac{\partial}{\partial x} (y + 2)

What is the integral of ((4x+13))/(x^2+3x-10) ?
The integral of ((4x+13))/(x^2+3x-10) is 2ln|4x^2+12x-40|-ln| 10/7+2/7 x|+ln|-4/7+2/7 x|+C