NCERT Class 12 · Ch 7 Exercise 7.4 · Q8
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.4 Question 8
Evaluate \(\int \frac{x^{2}}{\sqrt{a^{6} + x^{6}}}\, dx\).
💡 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≠0: (1/3)log|x³+√(a⁶+x⁶)|+C. For a=0,x≠0: sign(x)log|x|+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).
- u=x³, du=3x²dx; use ∫du/√(u²+a⁶). For a=0 the primitive is sign(x)log|x|.
- An antiderivative is F(x)=\(\frac{\log{\left(\left|{x^{3} + \sqrt{a^{6} + x^{6}}}\right| \right)}}{3}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{2}}{\sqrt{a^{6} + x^{6}}}\). 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 \(\int \frac{x - 1}{\sqrt{x^{2} - 1}}\, dx\).
Next Question →Evaluate \(\int \frac{\sec^{2}{\left(x…
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).