∫dx5sinx+2cosx+2 is equal to
(where C is integration constant)
A
ln∣∣∣5tanx2+2∣∣∣+C
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B
15ln∣∣∣5tanx2+2∣∣∣+C
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C
15ln∣∣∣5tanx2−2∣∣∣+C
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D
ln∣∣∣5tanx2−2∣∣∣+C
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Solution
The correct option is B15ln∣∣∣5tanx2+2∣∣∣+C I=∫dx5sinx+2cosx+2
Put tanx2=t ⇒sec2x2dx=2dt
Now, I=∫2dt1+t25⋅2t1+t2+2⋅1−t21+t2+2⎡⎢
⎢⎣∵sinx=2tanx21+tan2x2,cosx=1−tan2x21+tan2x2⎤⎥
⎥⎦⇒I=∫2dt10t+2−2t2+2+2t2⇒I=∫dt5t+2⇒I=15ln|5t+2|+C∴I=15ln∣∣∣5tanx2+2∣∣∣+C