NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 15

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Choose u=\(\log{\left(x \right)}\) and v=\(\frac{x^{3}}{3} + x\), so dv=\(x^{2} + 1\)dx and du=\(\frac{1}{x}\)dx.
3. Apply ∫u dv=uv−∫v du. The remaining integrand v·u′ is \(\frac{x^{2}}{3} + 1\). Integrate it and subtract from uv.
4. An antiderivative is F(x)=\(- \frac{x^{3}}{9} - x + \left(\frac{x^{3}}{3} + x\right) \log{\left(\left|{x}\right| \right)}\).
5. Differentiate this expression to check that F′(x)=\(\left(x^{2} + 1\right) \log{\left(x \right)}\). Add an arbitrary constant C on each domain component.