Questions for 1st Sem

Topic: Beta and Gamma Function

 Q1. Evaluate $\int_0^1 x^4 (1-\sqrt{x})dx$

Q2. Evaluate $\int_0^1 (1-x^3)^{-\frac{1}{2}}dx$

Q3. Show that $\int_0^{\frac{\pi}{2}} \sin^p \theta \cos^q  \theta d\theta=\frac{\Gamma(\frac{p+1}{2})\Gamma(\frac{q+1}{2})}{2 \Gamma(\frac{p+q+2}{2})}$

Q4. Show that $\int_0^{\frac{\pi}{2}} \sqrt{\cot \theta} d\theta =\frac{1}{2}\Gamma(\frac{1}{4})\Gamma(\frac{3}{4})$.

Q5. Show that $\Gamma(n)\Gamma(1-n)=\frac{\pi}{\sin n\pi}, 0<n<1$

Q6. Show that $\int_0^{\frac{\pi}{2}} \tan^p \theta d\theta =\frac{\pi}{2} \sec \frac{p\pi}{2}$

Q7. Express the follwing integrals in terms of Gamma Function

    a) $\int_0^1 \frac{dx}{\sqrt{1-x^4}}$                b) $\int_0^{\frac{\pi}{2}} \sqrt(\tan \theta)d\theta$

    c) $\int_0^{\infty}\frac{x^c}{c^x}dx$                d) $\int_0^{\infty} a^{-bx^2}dx$

    e) $\int_0^1 x^5 [log(1/x)]^3 dx$

Q8. Prove that $\Gamma(m)\Gamma(m+\frac{1}{2})=\frac{\sqrt{\pi}}{2^{2m-1}}\Gamma(2m)$