NCERT Class 12 · Ch 7 Exercise 7.4 · Q15
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 15
Evaluate \(\int \frac{1}{\sqrt{\left(- a + x\right) \left(- b + x\right)}}\, dx\).
💡 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: For a≠b: log|x−(a+b)/2+√((x−a)(x−b))|+C on each real open domain component. For a=b,x≠a: sign(x−a)log|x−a|+C.
- Work on each real open domain component; parameters are real, with a≠0 in Q8 and a≠b in Q15 (degenerate cases described below).
- Complete the square: (x−(a+b)/2)²−(a−b)²/4. If a=b, the primitive is sign(x−a)log|x−a|.
- An antiderivative is F(x)=\(\log{\left(\left|{\frac{a}{2} + \frac{b}{2} - x - \sqrt{\left(a - x\right) \left(b - x\right)}}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{\sqrt{\left(- a + x\right) \left(- b + x\right)}}\). 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 \(\int \frac{1}{- x^{2} + 3 x + 8}\, dx\).
Next Question →Evaluate \(\int \frac{4 x + 1}{\sqrt{2 x^{2} + x - 3}}\, dx\).
All Questions in Integrals — Exercise 7.4
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).