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Question

dxsin2x12cos2x+cosx sinx is equal to

A
17lntanx+3tanx4+C
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B
17lntanx3tanx+4+C
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C
17lntanx+3tanx+4+C
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D
17lntanx3tanx4+C
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Solution

The correct option is B 17lntanx3tanx+4+C
Solving the intregral of the form I=dxacos2x+bsin2x+cosx sinx
requires multiplying both numerator and denominator of the integrand by sec2x and substituting t=tanx
Here, multiplying the numerator and denominator of the integrand by sec2x we get,
I=sec2x dxsin2x sec2x+cosx sinx sec2x12cos2x sec2x I=sec2x dxtan2x+tanx12
Now, substituting t=tanx
we get, dt=sec2x dx
Substituting these back in the integral we get,
I=dtt2+t12I=dt(t3)(t+4)I=17(1t31t+4]dtI=17(ln(t3|ln(t+4|]+CI=17ln(t3t+4+C
Now, substituting back t=tanx
we get I=17ln(tanx3tanx+4+C
Thus, Option b. is the correct.

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