Question

Integrate (2cosx/Sin²x)dx ​

Answer 1

$\bold{ANSWER≈}$

I=integ.of 2.cos x.dx/(1-sinx)(1+sin^2x).

Let. sin x=p

cos x.dx=dp

I=integ.of 2.dp/(1-p)(1+p^2)

2/(1-p)(1+p^2)=A/(1-p)+(B.p+C)/(1+p^2)

2=A(1+p^2)+(B.p+C).(1-p)

2=(A+C)+(B-C).p+(A-B).p^2

A+C=2………..(1)

B-C=0…………..(2)

A-B=0…………….(3)

On adding eqn.(2) & (3)

A-C=0. or. A=C. or. A=C=1 with the help of eqn.(1).

On putting C=1 in eqn.(2). B=1.

I=integ. of [1/(1-p) + (p+1)/(1+p^2)].dp

I=integ.of [ 1/(1-p)+1/2. 2p/(1+p^2) + 1/(1+p^2)].dp

I= -log|1-p|+1/2.log|1+p^2|+tan^-1(p)+C

I=log√|1+p^2|/|1-p|+tan^-1(p)+C

On putting p= sin x

I= log √|1+sin^2x|/|1-sin x|. + tan^-1(sin x)+C. Answer.

Answer 2

Answer:

I=integ.of 2.cos x.dx/(1-sinx)(1+sin^2x).

Let. sin x=p

cos x.dx=dp

I=integ.of 2.dp/(1-p)(1+p^2)

2/(1-p)(1+p^2)=A/(1-p)+(B.p+C)/(1+p^2)

2=A(1+p^2)+(B.p+C).(1-p)

2=(A+C)+(B-C).p+(A-B).p^2

A+C=2………..(1)

B-C=0…………..(2)

A-B=0…………….(3)

On adding eqn.(2) & (3)

A-C=0. or. A=C. or. A=C=1 with the help of eqn.(1).

On putting C=1 in eqn.(2). B=1.

I=integ. of [1/(1-p) + (p+1)/(1+p^2)].dp

I=integ.of [ 1/(1-p)+1/2. 2p/(1+p^2) + 1/(1+p^2)].dp

I= -log|1-p|+1/2.log|1+p^2|+tan^-1(p)+C

I=log√|1+p^2|/|1-p|+tan^-1(p)+C

On putting p= sin x

I= log √|1+sin^2x|/|1-sin x|. + tan^-1(sin x)+C.

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