NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 9

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Choose u=\(\operatorname{acos}{\left(x \right)}\) and v=\(\frac{x^{2}}{2}\), so dv=\(x\)dx and du=\(- \frac{1}{\sqrt{1 - x^{2}}}\)dx.
3. Apply ∫u dv=uv−∫v du. The remaining integrand v·u′ is \(- \frac{x^{2}}{2 \sqrt{1 - x^{2}}}\). Integrate it and subtract from uv.
4. An antiderivative is F(x)=\(- \frac{x \sqrt{1 - x^{2}}}{4} + \frac{\left(2 x^{2} - 1\right) \operatorname{acos}{\left(x \right)}}{4}\).
5. Differentiate this expression to check that F′(x)=\(x \operatorname{acos}{\left(x \right)}\). Add an arbitrary constant C on each domain component.