Last updated on August 5th, 2025
In calculus, integration by parts is a technique used to integrate products of functions. The formula for integration of uv is derived from the product rule of differentiation and is essential in solving integrals involving products of functions. In this topic, we will learn the formula for the integration of uv.
Integration of uv is a method to integrate products of functions. Let’s learn the formula to calculate the integration of uv.
The integration of uv, also known as integration by parts, is derived from the product rule of differentiation.
It is given by the formula: ∫u dv = uv - ∫v du where u and v are differentiable functions of a variable. This formula helps integrate products of functions by differentiating one function and integrating another.
In math and real life, we use the integration of uv formula to solve complex integrals involving products of functions.
Here are some important aspects of integration of uv:
Students often find the integration of uv formula tricky and confusing.
Here are some tips and tricks to master it:
Integration of uv plays a major role in solving real-life problems involving complex integrals.
Here are some applications of the integration of uv formula:
Students make errors when applying the integration of uv formula. Here are some mistakes and the ways to avoid them, to master the technique.
Integrate x * e^x with respect to x.
The integral is x * e^x - e^x + C
Choose u = x and dv = ex dx.
Then, du = dx and v = ex
Using the formula, ∫x ex dx = x * ex - ∫ex dx = x * ex - ex + C
Integrate ln(x) with respect to x.
The integral is x * ln(x) - x + C
Choose u = ln(x) and dv = dx.
Then, du = (1/x) dx and v = x.
Using the formula, ∫ln(x) dx = x * ln(x) - ∫x * (1/x) dx = x * ln(x) - x + C
Integrate x^2 * sin(x) with respect to x.
The integral is -x^2 * cos(x) + 2∫x * cos(x) dx
Choose u = x2 and dv = sin(x) dx.
Then, du = 2x dx and v = -cos(x).
Using the formula, ∫x2 * sin(x) dx = -x2 * cos(x) + ∫2x * cos(x) dx
Integrate e^x * cos(x) with respect to x.
The integral is e^x * sin(x) + C
Choose u = ex and dv = cos(x) dx.
Then, du = ex dx and v = sin(x).
Using the formula, ∫ex * cos(x) dx = ex * sin(x) - ∫ex * sin(x) dx
Integrate x * arctan(x) with respect to x.
The integral is (1/2) * x^2 * arctan(x) - (1/2) * ∫x^2/(1+x^2) dx
Choose u = arctan(x) and dv = x dx.
Then, du = 1/(1+x2) dx and v = (1/2) * x2.
Using the formula, ∫x * arctan(x) dx = (1/2) * x2 * arctan(x) - (1/2) ∫x2/(1+x2) dx
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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