Class 12 Integrals · Short Answer2 Marksindefinite_integration
Class 12 Maths · Integrals · Short Answer #140
Evaluate \(\int \frac{x}{x \tan{\left(x \right)} + 1}\,dx\). Work on intervals with cos x≠0 and 1+x tan x≠0.
✅ Step-by-Step Solution
Final Answer: \(\log{\left(\left|{x \sin{\left(x \right)} + \cos{\left(x \right)}}\right| \right)}\) + C
- Let D=cos x+x sin x. Its derivative is D′=x cos x.
- Where the original integrand is defined, x/(1+x tan x)=x cos x/D=D′/D. Integrate as ln|D|.
- Therefore the required integral is \(\log{\left(\left|{x \sin{\left(x \right)} + \cos{\left(x \right)}}\right| \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let D=cos x+x sin x. Its derivative is D′=x cos x.
- [1 mark] Where the original integrand is defined, x/(1+x tan x)=x cos x/D=D′/D. Integrate as ln|D|.
- [0.5 marks] Therefore the required integral is \(\log{\left(\left|{x \sin{\left(x \right)} + \cos{\left(x \right)}}\right| \right)}\) + C.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Find one function f satisfying \(\int[(x-1)/x^2]e^x dx=e^x f(x)+C\)…
Next Question →Evaluate \(\int \frac{\left(x + 3\right) e^{x}}{\left(x +…
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).