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Please,please help me - Integral Calculus - JEE Main-5

Integrate \int \frac{dx}{\sin ^{2}x+2\sin x\cos x}

  • Option 1)

    \ln \frac{t+1}{t-1}+C

  • Option 2)

    \ln \frac{t+2}{t}+C

  • Option 3)

    \ln \frac{t}{t+2}+C

  • Option 4)

    none of these

 
Answers (1)
89 Views
V Vakul

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

 

 

Divide by \cos ^{2}x

\int \frac{dx\sec ^{2}x}{\tan ^{2}x+2\tan x}=\int \frac{dt}{\left ( t+1 \right )^{2}-1}=\ln \frac{t}{\left ( t+2 \right )}+C


Option 1)

\ln \frac{t+1}{t-1}+C

Option 2)

\ln \frac{t+2}{t}+C

Option 3)

\ln \frac{t}{t+2}+C

Option 4)

none of these

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