Integration by parts for Riemann integral

Integration by Parts for Riemann Integral Integration by parts is a technique used in calculus to evaluate definite integrals by breaking them down into smal...

Integration by Parts for Riemann Integral#

Integration by parts is a technique used in calculus to evaluate definite integrals by breaking them down into smaller, easier integrals. This method involves selecting two functions, often from the family of Riemann-integrable functions, and applying a formula to calculate the integral.

Basic Principle:

Let (u) and (dv) be two functions from the family of Riemann-integrable functions. Then the integral of (f(x)) with respect to (g(x)) is given by the following formula:

$\int_a^b f(x)g(x)dx = \left[u(x)\right]_a^b \int_a^b g(x)dx$

where (u(x)) and (v(x)) are functions of (x) that are continuously differentiable on the interval (a) to (b).

Applying the Formula:

To apply the formula, we choose (u(x) = f(x)) and (v(x) = g(x)). Then (du = f(x)dx) and (dv = g(x)dx), which are both continuous functions.

Substituting these values into the formula, we get:

$\begin{split}\int_a^b f(x)g(x)dx &= \left[f(x)\right]_a^b \int_a^b g(x)dx \\\ & = \left[f(x)\right]_a^b g(x)dx \\\ & = \left[f(x)\right]_a^b \left(\frac{1}{b}g(x)\right)_{a}^{b} \\\ & = \frac{1}{b} \int_a^b f(x)g(x)dx \end{split}$

Interpretation:

The result of the integration by parts formula tells us that the definite integral of (f(x)) with respect to (g(x)) can be calculated by evaluating the new integral at the endpoints of the original interval and subtracting the value at the starting point.

Examples:

Evaluate (\int_0^1 x^2e^{-x}dx), using (u = x^2) and (dv = e^{-x}dx). Then (u(0) = 0) and (v(x) = -e^{-x}), resulting in:

$\int_0^1 x^2e^{-x}dx = -e^{-x}\left[\frac{x^2}{2}\right]_0^1 = -\frac{e^{-1}}{2}$

Evaluate (\int_0^2 \frac{x}{x+1}dx), using (u = x) and (dv = \frac{1}{x+1}dx). Then (u(0) = 0) and (v(x) = x), resulting in:

$\int_0^2 \frac{x}{x+1}dx = [x]_0^2 = 2$

Benefits of Integration by Parts:

This method can be applied to integrals involving various functions from the Riemann family.
It simplifies the integration process by breaking it down into smaller, easier integrals.
The formula provides a clear and direct way to compute the integral.

Conclusion:

Integration by parts is a powerful technique in calculus that allows us to evaluate definite integrals by converting them into simpler integrals. By understanding the basic principles and applying the formula correctly, students can master this technique and apply it to solve various types of integration problems