Problem 82
The Beta function, which is important in many branches of mathematics, is defined as $$ B(\alpha, \beta)=\int_{0}^{1} x^{\alpha-1}(1-x)^{\beta-1} d x $$ with the condition that \(\alpha \geq 1\) and \(\beta \geq 1\). (a) Show by a change of variables that $$ B(\alpha, \beta)=\int_{0}^{1} x^{\beta-1}(1-x)^{\alpha-1} d x=B(\beta, \alpha) $$ (b) Integrate by parts to show that \(B(\alpha, \beta)=\frac{\alpha-1}{\beta} B(\alpha-1, \beta+1)=\frac{\beta-1}{\alpha} B(\alpha+1, \beta-1)\) (c) Assume now that \(\alpha=n\) and \(\beta=m\), and that \(n\) and \(m\) are positive integers. By using the result in part (b) repeatedly, show that $$ B(n, m)=\frac{(n-1) !(m-1) !}{(n+m-1) !} $$
Problem 83
. Suppose that \(f(t)\) has the property that \(f^{\prime}(a)=f^{\prime}(b)=0\) and that \(f(t)\) has two continuous derivatives. Use integration by parts to prove that \(\int_{a}^{b} f^{\prime \prime}(t) f(t) d t \leq 0 .\) Hint \(:\) Use integration by parts by differentiating \(f(t)\) and integrating \(f^{\prime \prime}(t) .\) This result has many applications in the field of applied mathematics.
Problem 84
Derive the formula $$ \int_{0}^{x}\left(\int_{0}^{t} f(z) d z\right) d t=\int_{0}^{x} f(t)(x-t) d t $$ using integration by parts.
Problem 86
If \(P_{n}(x)\) is a polynomial of degree \(n\), show that $$ \int e^{x} P_{n}(x) d x=e^{x} \sum_{j=0}^{n}(-1)^{j} \frac{d^{j} P_{n}(x)}{d x^{j}} $$
