JEEAdvanced
Using the Beta function (or Wallis reduction), evaluate I =…
Question
Using the Beta function (or Wallis reduction), evaluate I = \int_0^{\pi/2} \sin^6 x\,\cos^4 x\,dx. It equals \frac{m}{n}\pi with m/n in lowest terms. Find m+n.
✓ Verified answer: 515checked by our engine — not a guess
Step-by-step solution
Use \int_0^{\pi/2}\sin^{2a-1}x\cos^{2b-1}x\,dx = \frac12 B(a,b) = \frac{\Gamma(a)\Gamma(b)}{2\Gamma(a+b)}. Here 2a-1=6\Rightarrow a=7/2, 2b-1=4\Rightarrow b=5/2, a+b=6. So I = \frac{\Gamma(7/2)\Gamma(5/2)}{2\Gamma(6)}. With \Gamma(7/2)=\frac{15\sqrt\pi}{8}, \Gamma(5/2)=\frac{3\sqrt\pi}{4}, \Gamma(6)=120: numerator = \frac{15\sqrt\pi}{8}\cdot\frac{3\sqrt\pi}{4} = \frac{45\pi}{32}; divide by 2\cdot120=240: I = \frac{45\pi}{32\cdot240} = \frac{45\pi}{7680} = \frac{3\pi}{512}. So m=3, n=512 and m+n = 515.
Final answer515
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Definite Integrals solutions
Evaluate I = \int_0^1 \frac{x^4(1-x)^4}{1+x^2}\,dx. It can be written…JEE · MathEvaluate I = \int_0^{\pi/2} \ln^2(\sin x)\,dx and give its value…JEE · MathUsing the symmetry x\to\pi-x, evaluate I = \int_0^{\pi} x\,\sin^4…JEE · MathEvaluate I = \int_0^{2\pi} \frac{dx}{(5-4\cos x)^2}. The result is…JEE · MathBy differentiating under the integral sign, evaluate I =…JEE · MathUsing the reduction formula for powers of tangent, evaluate I =…JEE · MathEvaluate I = \int_0^{1} \frac{(\ln x)^2}{1+x^2}\,dx. It equals…JEE · MathEvaluate I = \int_0^{\pi} \ln\!\big(5 - 4\cos x\big)\,dx and give its…JEE · Math