The value of √2∫sinxdxsin(x−π4) is
(where C is constant of integration)
A
x+ln∣∣∣cos(x−π4)∣∣∣+C
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B
x−ln∣∣∣sin(x−π4)∣∣∣+C
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C
x+ln∣∣∣sin(x−π4)∣∣∣+C
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D
x−ln∣∣∣cos(x−π4)∣∣∣+C
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Solution
The correct option is Cx+ln∣∣∣sin(x−π4)∣∣∣+C Let I=√2∫sinxdxsin(x−π4)
Put x−π4=t ⇒dx=dt ∴I=√2∫sin(t+π4)sintdt ⇒I=√2√2∫(sint+costsint)dt ⇒I=∫(1+cott)dt ⇒I=t+ln|sint|+c1 ⇒I=x−π4+ln∣∣∣sin(x−π4)∣∣∣+c1 ∴I=x+ln∣∣∣sin(x−π4)∣∣∣+C (whereC=c1−π4)