Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #85
Evaluate \(\int \log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}^{2}}\,dx\). Work for x>1.
✅ Step-by-Step Solution
Final Answer: \(x \log{\left(\log{\left(x \right)} \right)} - \frac{x}{\log{\left(x \right)}}\) + C
- The real domain is x>1. Let u=ln x, so dx=e^u du.
- The integral becomes ∫e^u[ln u+u^(−2)]du. With g(u)=ln u−1/u, one has g+g′=ln u+u^(−2).
- Use d[e^u g(u)]/du=e^u(g+g′); substitute u=ln x to get xln(ln x)−x/ln x.
- Therefore the required integral is \(x \log{\left(\log{\left(x \right)} \right)} - \frac{x}{\log{\left(x \right)}}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The real domain is x>1. Let u=ln x, so dx=e^u du.
- [1 mark] The integral becomes ∫e^u[ln u+u^(−2)]du. With g(u)=ln u−1/u, one has g+g′=ln u+u^(−2).
- [1 mark] Use d[e^u g(u)]/du=e^u(g+g′); substitute u=ln x to get xln(ln x)−x/ln x.
- [0.5 marks] Therefore the required integral is \(x \log{\left(\log{\left(x \right)} \right)} - \frac{x}{\log{\left(x \right)}}\) + C.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int - \frac{1}{\sin{\left(a - x \right)} \cos{\left(b…
Next Question →Evaluate \(\int_{-1}^2|x^3-x|dx\).
More Practice Questions in Integrals
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).