How do you integrate $ $ $ \int {\dfrac{{5x + 10}}{{{x^2} + 2x - 35}}dx} $ using partial fraction?

Hint: To integrate a function using partial fraction method we first express the function in partial ratio notation i.e. $ \dfrac{{p(x)}}{{q(x)}} $ where the value of $ p(x) $ should not be equal to zero.

Complete step by step solution:
First of all perform the decomposition into the partial fractions
 $ \dfrac{{5x - 10}}{{{x^2} + 2x - 35}} = \dfrac{{5x - 10}}{{(x + 7)(x - 5)}} $
 $ \dfrac{{5x - 10}}{{{x^2} + 2x - 35}} = \dfrac{P}{{x + 7}} + \dfrac{Q}{{x - 5}} $
 $ = \dfrac{{P(x - 5) + Q(x + 7)}}{{(x + 7)(x - 5)}} $
As the denominators are same, then on comparing the numerators we get
 $ 5x - 10 = P(x - 5) + Q(x + 7) $
Let
  $
  x = - 7 \\
   \Rightarrow - 45 = - 12P \\
   \Rightarrow P = \dfrac{{45}}{{12}} = \dfrac{{15}}{4} \;
  $
Further, let $ x = 5 $
 $
   \Rightarrow 15 = 12Q \\
   \Rightarrow Q = \dfrac{{15}}{{12}} = \dfrac{5}{4} \;
  $
Therefore, we have
 $ \dfrac{{5x - 10}}{{{x^2} + 2x - 35}} = \dfrac{{\dfrac{{15}}{4}}}{{x + 7}} + \dfrac{{\dfrac{5}{4}}}{{x - 5}} $
Now, taking integral on both sides with respect to x we get
 $ \int {\dfrac{{5x - 10}}{{{x^2} + 2x - 35}}dx} = \int {\dfrac{{\dfrac{{15}}{4}}}{{x + 7}}} dx + \int {\dfrac{{\dfrac{5}{4}}}{{x - 5}}} dx $
As we know that $ \int {\dfrac{1}{x}dx} = \ln x + c $ we have
\[\int {\dfrac{{5x - 10}}{{{x^2} + 2x - 35}}dx} = \dfrac{{15}}{4}\ln (x + 7) + \dfrac{5}{4}\ln (x - 5) + c\]
Hence this is the required answer.
So, the correct answer is “\[\int {\dfrac{{5x - 10}}{{{x^2} + 2x - 35}}dx} = \dfrac{{15}}{4}\ln (x + 7) + \dfrac{5}{4}\ln (x - 5) + c\] ”.

Note: here are two forms of partial fractions which are described below:
I.Proper Partial Fractions-When the degree of the denominator is more than the degree of the numerator, the function is known as a proper partial fraction.
II.Improper Partial Fractions- When the degree of the denominator is less than the degree of numerator then the fraction is known as Improper Partial Fraction. Thus, the fraction can be simplified into parts and can be integrated easily.