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The value of is \dpi{100} \sqrt{2}\int \frac{\sin x\, dx}{\sin \left ( x-\frac{\pi }{4} \right )}

  • Option 1)

    x-\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c

  • Option 2)

    x+\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c

  • Option 3)

    x-\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c

  • Option 4)

    x+\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c

 

Answers (1)

best_answer

As learnt in the concept

Integrals for Trigonometric functions -

\frac{\mathrm{d} }{\mathrm{d} x}\left ( -cos x \right ) =sinx

\therefore \int sinxdx=-cosx+c

-

 

 \sqrt{2}\int \frac{sinxdx}{sin(x-\frac{\pi}{4})}

=\sqrt{2}\int \frac{sin(x-\pi/4 +\pi/4)dx}{sin(x-\frac{\pi}{4})}

=\sqrt{2}\int \left (cos\frac{\pi }{4} + cot\left (x-\frac{\pi }{4} \right )sin\frac{\pi }{4} \right )dx

=\sqrt{2}\int \frac{1 }{\sqrt{2}}dx +\sqrt{2}\int \frac{1}{\sqrt{2}}cot\left (x-\frac{\pi }{4} \right )dx

= x+ \int cot\left (x-\frac{\pi }{4} \right )dx

= x+ log \left | sin\left (x-\frac{\pi}{4} \right ) \right |+C


Option 1)

x-\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c

Option 2)

x+\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c

Option 3)

x-\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c

Option 4)

x+\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c

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