5x²+3/x²(x²-2) dx =

Correct Answer is : 1342log⁡∣x−2x+2∣+32x+c{13}{4{2}}≤ft|{x-{2}}{x+{2}}|+3/2x+c4213logx+2x−2+2x3+c

{13}{4{2}} ≤ft| {x+{2}}{x-{2}} | + 3/2x + c

{13}{2{2}} ≤ft| {{2}-x}{{2}+x} | + 3/2x + c

{5}{3{2}}≤ft|{x+{2}}{x-{2}}|+3/5x+c

{13}{4{2}}≤ft|{x-{2}}{x+{2}}|+3/2x+c

Correct Answer is : 1342log⁡∣x−2x+2∣+32x+c{13}{4{2}}≤ft|{x-{2}}{x+{2}}|+3/2x+c4213logx+2x−2+2x3+c

Test Index

AP EAMCET Engineering 26 Apr 2017 Shift 1 Paper

Section: Mathematics

Question : 74 of 160

Marks: +1, -0

\int \frac{5x^2+3}{x^2(x^2-2)} dx =

$\frac{13}{4\sqrt{2}} \log \left| \frac{x+\sqrt{2}}{x-\sqrt{2}} \right| + \frac{3}{2x} + c$
$\frac{13}{2\sqrt{2}} \log \left| \frac{\sqrt{2}-x}{\sqrt{2}+x} \right| + \frac{3}{2x} + c$
$\frac{5}{3\sqrt{2}}\log\left|\frac{x+\sqrt{2}}{x-\sqrt{2}}\right|+\frac{3}{5}x+c$
$\frac{13}{4\sqrt{2}}\log\left|\frac{x-\sqrt{2}}{x+\sqrt{2}}\right|+\frac{3}{2x}+c$

Solution:

Consider the integral.

I = \int \frac{5x^2+3}{x^2(x^2-2)} dx

Let

x^2 = y

\frac{5x^2+3}{x^2(x^2-2)} = \frac{5y+3}{y(y-2)}

Thus,

\frac{5y+3}{y(y-2)} = \frac{A}{y} + \frac{B}{y-2}

5y+3 = A(y-2) + By

Compare the coefficient of like terms.

A+B = 5

2A = -3

So,

A = -\frac{3}{2}

B = \frac{13}{2}

Hence,

\frac{5x^2+3}{x^2(x^2-2)} = -\frac{3}{2} \cdot \frac{1}{x^2} + \frac{13}{2} \cdot \frac{1}{x^2-2}

Therefore,

I = \int \left( -\frac{3}{2} \frac{1}{x^2} + \frac{13}{2} \frac{1}{x^2-2} \right) dx

= -\frac{3}{2} \left( -\frac{1}{x} \right) + \frac{13}{2} \left( \frac{1}{2\sqrt{2}} \right) \log \left| \frac{x-\sqrt{2}}{x+\sqrt{2}} \right| + c

= \frac{3}{2x} + \frac{13}{4\sqrt{2}} \log \left| \frac{x-\sqrt{2}}{x+\sqrt{2}} \right| + c

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