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NCERT Class 12 · Ch 7 Exercise 7.6 · Q11

NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 11

Evaluate \(\int \frac{x \operatorname{acos}{\left(x \right)}}{\sqrt{1 - x^{2}}}\, dx\).

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

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

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All Questions in Integrals — Exercise 7.6

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