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Question

Match the columns by choosing the correct answer from B for A.

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Solution

A.
cotxdx=?
Let I=cotxdx
I=cosxsinxdx(1)
Put sinx=t(2)
cosxdx=dt(3)
Put (3) in (1)
I=dtt=log|t|+c
I=log|sinx|+c
cotxdx=log|sinx|+c

B.
tanxdx=?
Let I=tanxdx
I=sinxcosxdx(1)
Put cosx=t(2)
sinxdx=dt(3)
Put (3) in (1)
I=dtt=log|t|+c
I=log|cosx|+c
I=log1cosx+c
I=log|secx|+c
tanxdx=log|secx|+c

C.
11+cos2xdx=?
Using, cos2x=2cos2x1
I=11+cos2xdx=11+2cos2x1dx
I=12cos2xdx
=12sec2xdx
Using sec2xdx=tanx
I=tanx2+c

D.
(1+tan2x)dx=?
As, we know that 1+tan2x=sec2x
(1+tan2x)dx=(sec2x)dx
Using, ddxtanx=sec2x
We have,
(sec2x)dx=tanx+c
(1+tan2x)dx=tanx+c

E.
cscxdx=?
I=cscxdx
=cscx(cscx+cotx)cscx+cotxdx(1)
cscx+cotx=t(2)
cscx(cscx+cotx)dx=dt
Put (3) in (1)
I=dtt
I=log|t|+c=log|cscx+cotx|+c
log|cscx+cotx|+c=log1sinx+cosxsinx+c
=log1+cosxsinx+c
=log∣ ∣ ∣ ∣1+(2cos2x21)sinx∣ ∣ ∣ ∣+c
=log∣ ∣ ∣2cos2x22sinx2cosx2∣ ∣ ∣+c
=log∣ ∣ ∣2cosx2cosx22sinx2cosx2∣ ∣ ∣+c
=log∣ ∣ ∣cosx2sinx2∣ ∣ ∣+c
=logcotx2+c
=log∣ ∣ ∣1cotx2∣ ∣ ∣+c
=logtanx2+c

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