NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 8

✅ Step-by-Step Solution

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