NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q15
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 15
Evaluate \(\int \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx\). Original logarithm domain: sin x>0.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(- \frac{\cos^{4}{\left(x \right)}}{4}\) + C.
- Work on each open interval where the original integrand is real and defined.
- The real log requires sin x>0. Then e^(log sin x)=sin x; put u=cos x.
- An antiderivative is F(x)=\(- \frac{\cos^{4}{\left(x \right)}}{4}\).
- Differentiate this expression to check that F′(x)=\(\sin{\left(x \right)} \cos^{3}{\left(x \right)}\). Add an arbitrary constant C on each domain component.
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