Class 12 Integrals · Short Answer2 Marksindefinite_integration
Class 12 Maths · Integrals · Short Answer #45
Evaluate \(\int \frac{1}{x^{2} - 6 x + 13}\,dx\). Work on each connected open interval where the real integrand is defined.
✅ Step-by-Step Solution
Final Answer: \(\frac{\operatorname{atan}{\left(\frac{x}{2} - \frac{3}{2} \right)}}{2}\) + C
- Complete the square: x²−6x+13=(x−3)²+4.
- Use u=(x−3)/2, dx=2du, giving (1/2)∫du/(1+u²)=(1/2)arctan u+C.
- Therefore the required integral is \(\frac{\operatorname{atan}{\left(\frac{x}{2} - \frac{3}{2} \right)}}{2}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Complete the square: x²−6x+13=(x−3)²+4.
- [1 mark] Use u=(x−3)/2, dx=2du, giving (1/2)∫du/(1+u²)=(1/2)arctan u+C.
- [0.5 marks] Therefore the required integral is \(\frac{\operatorname{atan}{\left(\frac{x}{2} - \frac{3}{2} \right)}}{2}\) + C.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int \frac{1}{\sqrt{- x^{2} + 4 x}}\,dx\). Work for 0<x<4.
Next Question →Evaluate \(\int_0^1x^2e^x dx\).
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).