Applications of Integration by Parts

Integration by Parts has various applications in integral calculus it is used to find the integration of the function where normal integration techniques fail. We can easily find the integration of inverse and logarithmic functions using the integration by parts concept.

We will find the Integration of the Logarithmic function and Arctan function using integration by part rule,

Integration of Logarithmic Function (log x)

Integration of Inverse Logarithmic Function (log x) is achieved using Integration by part formula. The integration is discussed below,

∫ logx.dx = ∫ logx.1.dx

Taking log x as the first function and 1 as the second function.

Using ∫u.v dx = u ∫ v d(x) – ∫ [u’ {∫v dx} dx] dx

⇒ ∫ logx.1.dx =  logx. ∫1.dx – ∫ ((logx)’.∫ 1.dx).dx

⇒ ∫ logx.1.dx = logx.x -∫ (1/x .x).dx

⇒ ∫ logx.1.dx = xlogx – ∫ 1.dx

⇒ ∫ logx.dx = x logx – x + C

Which is the required integration of logarithmic function.

Integration of Inverse Trigonometric Function (tan^-1 x)

Integration of Inverse Trigonometric Functions (tan^-1 x) is achieved using Integration by part formula. The integration is discussed below,

∫ tan^-1x.dx = ∫tan^-1x.1.dx

Taking tan^-1 x as the first function and 1 as the second function.

Using ∫u.v dx = u ∫ v d(x) – ∫ [u’ {∫v dx} dx] dx

⇒ ∫tan^-1x.1.dx = tan^-1x.∫1.dx – ∫((tan^-1x)’.∫ 1.dx).dx

⇒ ∫tan^-1x.1.dx = tan^-1x. x – ∫(1/(1 + x²).x).dx

⇒ ∫tan^-1x.1.dx = x. tan^-1x – ∫ 2x/(2(1 + x²)).dx

⇒ ∫tan^-1x.dx = x. tan^-1x – ½.log(1 + x2) + C

Which is the required integration of Inverse Trigonometric Function.

Real-life Applications of Partial Integration

Some of the common real life application of partial integration are:

• Finding Antiderivatives
  • In engineering and physics, partial integration is used to find antiderivatives of functions that represent physical quantities. For example, in mechanics, it’s used to derive equations of motion from the equations of force and acceleration.
• Wallis Product
  • The Wallis product, an infinite product representation of pi, can be derived using partial integration techniques. This product has applications in fields such as number theory, probability theory, and signal processing.
• Gamma Function Identity
  • The gamma function, which extends the factorial function to complex numbers, has various applications in mathematics, physics, and engineering. Partial integration is used to prove identities involving the gamma function, which are crucial in areas like probability theory, statistical mechanics, and quantum mechanics.
• Use in Harmonic Analysis
  • Partial integration plays a significant role in harmonic analysis, particularly in Fourier analysis. It is used to derive properties of Fourier transforms, such as the convolution theorem and properties of Fourier series. These results are applied in fields like signal processing, image analysis, and telecommunications.

Integration by Parts: Integration by parts is a technique used in calculus to find the integral of the product of two functions. It’s essentially a reversal of the product rule for differentiation.

Integrating a function is not always easy sometimes we have to integrate a function that is the multiple of two or more functions in this case if we have to find the integration we have to use integration by part concept, which uses two products of two functions and tells us how to find their integration.

Now let’s learn about Integration by parts, its formula, derivation, and others in detail in this article.