NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 10

✅ Step-by-Step Solution

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