Class 12 Maths · Integrals · Long Answer #65

Question

Evaluate $\int \frac{1}{\sqrt{\sin^{3}{\left(x \right)} \cos{\left(a - x \right)}}}\,dx$. Work on real open intervals where sin³x cos(x−a)>0, assuming real a with cos a≠0.

Accepted Answer

Final Answer: $- \frac{2 \sqrt{\sin{\left(a \right)} + \cos{\left(a \right)} \cot{\left(x \right)}}}{\cos{\left(a \right)}}$ + C Step 1: Factor sin³x cos(x−a)=sin⁴x[sin a+cos a cot x]. For positive radicand, the integrand is csc²x/√(sin a+cos a cot x). Step 2: Let u=√(sin a+cos a cot x). Then du=−cos a·csc²x dx/(2u). Step 3: Thus the integral is −(2/cos a)∫du=−2u/cos a+C; substitute back. Step 4: Therefore the required integral is $- \frac{2 \sqrt{\sin{\left(a \right)} + \cos{\left(a \right)} \cot{\left(x \right)}}}{\cos{\left(a \right)}}$ + C.

Class 12 Maths · Integrals · Long Answer #65

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