Question

Find \(\int_{0}^{1} x^{p}(1-x)^{q} d x\) (p and q positive integers).

Short answer

Question: Compute the definite integral of the function \(x^p(1-x)^q\) in the interval \([0, 1]\), where both \(p\) and \(q\) are positive integers. Short Answer: The definite integral can be expressed as the following after applying the integration by parts technique: $$\int_{0}^{1} x^p (1-x)^q dx = \left[x^p\left(-\frac{(1-x)^{q+1}}{q+1}\right) - \int\left(-\frac{(1-x)^{q+1}}{q+1}\right)p x^{p-1} dx\right]_{0}^{1}$$ Further simplification or assumptions about \(p\) and \(q\) may be necessary to evaluate this integral in closed form.

Step-by-step solution

Apply integration by parts formula

Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Assign \(u\) and \(dv\) as follows: - Choose \(u = x^p\) such that \(du = p x^{p-1} dx\) - Choose \(dv = (1-x)^q dx\) such that \(v = \int (1 - x)^q dx\)

Compute \(du\) and \(v\)

Compute the derivatives and integrals associated with \(u\) and \(dv\): - \(du = p x^{p-1} dx\) - To find \(v\), we use substitution: let \(t = 1 - x\). Then \(-dt = dx\), and we get: $$v = \int -(1 - x)^q dx = -\int t^q dt = -\frac{t^{q+1}}{q+1} + C = -\frac{(1-x)^{q+1}}{q+1} + C$$

Apply the integration by parts formula

Substitute \(u\), \(du\), \(v\), and \(dv\) into the integration by parts formula: $$\int x^p (1-x)^q dx = x^p\left(-\frac{(1-x)^{q+1}}{q+1} + C\right) - \int\left(-\frac{(1-x)^{q+1}}{q+1} + C\right)p x^{p-1} dx$$ Since we are dealing with definite integrals, we can ignore the constant C. $$\int x^p (1-x)^q dx = x^p\left(-\frac{(1-x)^{q+1}}{q+1}\right) - \int\left(-\frac{(1-x)^{q+1}}{q+1}\right)p x^{p-1} dx$$

Evaluate the definite integral

Now, we need to evaluate the definite integral between the bounds \(0\) and \(1\): $$\int_{0}^{1} x^p (1-x)^q dx = \left[x^p\left(-\frac{(1-x)^{q+1}}{q+1}\right) - \int\left(-\frac{(1-x)^{q+1}}{q+1}\right)p x^{p-1} dx\right]_{0}^{1}$$ By evaluating the expression at \(x = 1\) and \(x = 0\), we have the final result as a series of expressions involving \(p\) and \(q\). This integral may still be quite difficult to evaluate in closed form without additional simplifications or assumptions about \(p\) and \(q\). However, the demonstration of applying integration by parts technique is the primary objective of this problem.