NCERT Class 12 · Ch 7 Exercise 7.6 · Q13
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 13
Evaluate \(\int \operatorname{atan}{\left(x \right)}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(x \operatorname{atan}{\left(x \right)} - \frac{\log{\left(x^{2} + 1 \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Choose u=\(\operatorname{atan}{\left(x \right)}\) and v=\(x\), so dv=\(1\)dx and du=\(\frac{1}{x^{2} + 1}\)dx.
- Apply ∫u dv=uv−∫v du. The remaining integrand v·u′ is \(\frac{x}{x^{2} + 1}\). Integrate it and subtract from uv.
- An antiderivative is F(x)=\(x \operatorname{atan}{\left(x \right)} - \frac{\log{\left(x^{2} + 1 \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\operatorname{atan}{\left(x \right)}\). Add an arbitrary constant C on each domain component.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int x \sec^{2}{\left(x \right)}\, dx\).
Next Question →Evaluate \(\int x \log{\left(x \right)}^{2}\, dx\).
All Questions in Integrals — Exercise 7.6
Related Questions in Integrals
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int x^2e^{x^3}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(1+x)\sec^2(xe^x)\,dx\) on an interval where defined.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(\cos x-\sin x)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int dx/(\sin^2x\cos^2x)\) on an interval avoiding its poles.
Class 12 Integrals · Mcq Single Correct
If \(\int\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln|4e^x+5e^{-x}|+C\), find a,b.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).