Question

\(∫\dfrac{x+5}{x^{2}+1}dx=\)

• A. \( 3ln|x-1|-2ln|x+1|+C\)
• B. \(2ln|x-1|-3ln|x+1|+C\)
• C. \(ln|x-2|-ln|x+1|+C\)
• D. \( ln|x+2|+ln|x+1|+C\)
• E. \(2ln|x-1|+3ln|x-1|+C\)

Answer 1

The Correct Option is A

\(∫\dfrac{x+5}{x^{2}+1}dx=\)

So to solve this question, we can again use partial fraction decomposition.

Step 1: 

Factorize the denominator. \((x^2 - 1)\) can be factored as\((x - 1)(x + 1).\)

Step 2:

 Partial fraction decomposition. The expression \(\dfrac{x+5}{x^2 - 1}\) can be rewritten as the sum of two fractions with unknown constants \(A\) and \(B\): 

\(\dfrac{x+5}{x^2 - 1}  = \dfrac{A}{x +1}+\dfrac{B}{x-1}\)

Step 3: 

Now to find the values of A and B, we need to find a common denominator, which is \((x- 1)(x + 1)\), and then equate the numerators: 

\(x + 5 = A(x+ 1) + B(x- 1)\)

Now, solve for A and B by comparing coefficients: \(A + B = 1\)

 (by comparing the coefficients of \(x\)) \(A - B = 5\)  ⇢(by comparing the constant terms)

Adding the two equations: \(2A = 6\)

                                            ⇒\(A=3\)

Substituting the value of  \(A\) one of the equations to find we get 

\(B = -2\)

Step 4: 

Now we can re-write the parent expression as,

\(∫\dfrac{x + 5}{x^2 - 1} dx = ∫(\dfrac{A}{x - 1}+\dfrac{B}{x+ 1}) dx\)

\(=∫(\dfrac{3}{x-1}) - (\dfrac{2}{x+1})dx\)

\(=∫(\dfrac{3}{x-1}) - (\dfrac{2}{x+1})dx\)

\(=∫(\dfrac{3}{x-1})dx - ∫(\dfrac{2}{x+1})dx\)

\(= 3ln|x-1|-2ln|x+1|+C\) (Ans.)

Answer 2

\(∫\dfrac{x+5}{x^{2}+1}dx\)

So to solve this question, we can use partial fraction decomposition.

Step 1 :- $\int (\frac{3}{x-1} + \frac{-2}{x+1}) dx $

Step 2 :- $\int \frac{3}{x-1} dx+ \int \frac{-2}{x+1} dx $

Step 3 :- $3ln|x-1| + \int \frac{-2}{x+1} dx $

Step 4 :- $3ln|x-1| - 2 ln |x+1| +C$

So, the correct option is (A) : $3ln|x-1| - 2 ln |x+1| +C$