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Question

# If ∫(cos3x+cos5x)(sin2x+sin4x)dx=Asinx+B cosec x+Ctan−1(sinx)+K, where A,B,C are fixed constants and K is constant of integration, then the value of A+B−C is equal to

A
7
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B
3
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C
5
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D
4
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Solution

## The correct option is C 5Let I=∫(cos3x+cos5x)(sin2x+sin4x)dx ⇒I=∫(cos2x+cos4x)(sin2x+sin4x)cosxdx Put sinx=t ⇒cosxdx=dt ∴I=∫(1−t2)+(1+t4−2t2)t2+t4dt ⇒I=∫t4−3t2+2t2(t2+1)dt ⇒I=∫[1+(2−4t2)t2(t2+1)]dt ⇒I=∫1⋅dt+2∫(t2+1)−t2t2(t2+1)dt−4∫1t2+1dt ⇒I=∫1⋅dt+2∫1t2dt−6∫1t2+1 dt ⇒I=t−2t−6tan−1(t)+K ⇒I=sinx−2sinx−6tan−1(sinx)+K ⇒A=1,B=−2 and C=−6 ∴A+B−C=5

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