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  • Need clarity, kindly explain! - Integral Calculus - JEE Main-7

Find the integral \int \frac{\left ( x+2 \right )dx}{\left (x^{2}+4x+3 \right )}

  • Option 1)

    \ln \left ( x^{2}+4x+3 \right )+C

  • Option 2)

    \frac{1}{2}\ln \left ( x^{2}+4x+3 \right )+C

  • Option 3)

    2\ln \left ( x^{2}+4x+3 \right )+C

  • Option 4)

    none of these

 

Answers (1)

As we learned,

 

Type of Integration by perfect square -

The integral in the form of :

(i)    \int \frac{px+r}{ax^{2}+bx+c}dx     (ii) \int \frac{px+r}{\sqrt{{ax^{2}+bx+c}}}dx

(iii) \int (px+r){\sqrt{{ax^{2}+bx+c}}}dx  

\therefore \int \frac{px+r}{ax^{2}+bx+c}dx=A\frac{\ d.c.\ of\left ( ax^{2}+bx+c \right )}{ax^{2}+bx+c}+B\cdot \frac{1}{ax^{2}+bx+c}  and find integrals by standard formulae

 

- wherein

Working rule.

Let     px+r=A\frac{\mathrm{d} }{\mathrm{d} x}(ax^{2}+bx+c)+B

Find A and B by comparing

 

 

I=\frac{1}{2}\int \frac{\left ( 2x+4 \right )dx}{x^{2}+4x+3}

Put x^{2}+4x+3=t

\Rightarrow \: \left ( 2x+4 \right )dx=dt

I=\frac{1}{2}\ln \left ( x^{2}+4x+3 \right )+C


Option 1)

\ln \left ( x^{2}+4x+3 \right )+C

Option 2)

\frac{1}{2}\ln \left ( x^{2}+4x+3 \right )+C

Option 3)

2\ln \left ( x^{2}+4x+3 \right )+C

Option 4)

none of these

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subam

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