NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q13
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 13
Evaluate \(\int \frac{e^{x}}{\left(e^{x} + 1\right) \left(e^{x} + 2\right)}\, dx\).
💡 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: \(\log{\left(\frac{e^{x} + 1}{e^{x} + 2} \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Put u=eˣ, then split 1/[(u+1)(u+2)] into 1/(u+1)−1/(u+2).
- An antiderivative is F(x)=\(\log{\left(\frac{e^{x} + 1}{e^{x} + 2} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{e^{x}}{\left(e^{x} + 1\right) \left(e^{x} + 2\right)}\). 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 \(\int \frac{x^{3}}{\sqrt{1 - x^{8}}}\, dx\).
Next Question →Evaluate \(\int \frac{1}{\left(x^{2} + 1\right) \left(x^{2} +…
All Questions in Integrals — Miscellaneous Exercise
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).