NCERT Class 12 · Ch 7 Exercise 7.3 · Q4
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 4
Evaluate \(\int \sin^{3}{\left(2 x + 1 \right)}\, dx\).
💡 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: \(\frac{\cos^{3}{\left(2 x + 1 \right)}}{6} - \frac{\cos{\left(2 x + 1 \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Write sin³u=sin u(1−cos²u), then t=cos u and du=2dx.
- An antiderivative is F(x)=\(\frac{\cos^{3}{\left(2 x + 1 \right)}}{6} - \frac{\cos{\left(2 x + 1 \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\sin^{3}{\left(2 x + 1 \right)}\). Add an arbitrary constant C on each domain component.
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