Question

Evaluate the integral: \[ \int \frac{\sin x}{\sin\left(x-\frac{\pi}{4}\right)}\,dx \]

• A. \( \dfrac{1}{\sqrt{2}}\left[x+\log\left|\sin\left(x-\frac{\pi}{4}\right)\right|\right]+C \)
• B. \( x+\log\left|\sin\left(x-\frac{\pi}{4}\right)\right|+C \)
• C. \( x-\log\left|\sin\left(x-\frac{\pi}{4}\right)\right|+C \)
• D. \( \dfrac{1}{\sqrt{2}}\left[x-\log\left|\sin\left(x-\frac{\pi}{4}\right)\right|\right]+C \)

Hint:

Whenever trigonometric expressions involve shifted angles, convert them using standard identities before integration.

Answer 1

The Correct Option is A

Step 1: Use sine subtraction identity.
\[ \sin\left(x-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}(\sin x-\cos x) \] Step 2: Rewrite the integrand.
\[ \frac{\sin x}{\sin\left(x-\frac{\pi}{4}\right)} = \frac{\sqrt{2}\sin x}{\sin x-\cos x} \] Step 3: Split the expression.
\[ \frac{\sin x}{\sin x-\cos x} =1+\frac{\cos x}{\sin x-\cos x} \] Step 4: Integrate termwise.
\[ \int \frac{\sin x}{\sin\left(x-\frac{\pi}{4}\right)}dx =\frac{1}{\sqrt{2}}\left[x+\log\left|\sin\left(x-\frac{\pi}{4}\right)\right|\right]+C \]