Integration by Parts: Overview, Questions, Preparation

Integration by parts refers to a unique integration method, which is extremely useful while multiplying two functions together. This method is beneficial to find the integrals by reducing the terms into their standard forms. For instance, to find the integration of x sin x, the integration method is used. The utility of this integration method is beneficial in many ways, as well. The rule of this method is:
∫u v dx = u∫v dx −∫u' (∫v dx) dx.
Here, 
u = the function u (x),
v = the function v (x),
u' =  the derivative of function u (x).

Integration by parts typically uses the Ilate Rule, which helps to find the first and second function in the same method.
In integration by parts, when the multiple of two functions is given, apply the particular formula. The integral of these two functions are taken by considering them as the first and second function. This arrangement is known as Ilate Rule. 

The first integration by parts begin with
d (u v) = u d v + v d u.

Integrating on both the sides,
∫ d (u v) = u v = ∫ u d v + ∫ v d u.

Rearranging the above equation gives
∫ u d v = u v - ∫ v d u.

Definite integral

An integral having upper and lower limits is known as a definite integral. On a real plane, x is limited to lie. The other name of the definite integral is Riemann integral.

Indefinite integral

The indefinite integral is defined without any upper or lower limits, and they are represented as- ∫f(x)dx = F(x) + C

 Where, C = constant

Function f(x) is integrated

Integration by parts with examination point of view 

This covers introductory aspects in class 10th and class 11th and a deep understanding in class 12th around the topic: integration by parts. When the integrand function is represented as a product of two or more functions, integrating the given function can be done using the integration by parts rule. 

Integration by parts comes under section “calculus”. In class X, its weightage is nominal marks. In class XI, its weightage is 5 marks. In class XII, its weightage is 35 marks.

1. Integrate  ∫ x sinX.dx

Solution. Let,

u = x, dv= sinX.dx

du = (1) dx, and v= - cosX

∫ x sinX.dx = x (-cosX) - ∫ (- cosX) dx

= - x cosX + ∫ CosX.dx

= - x cosX + sinX + C

2. Integrate  ∫ x In x.dx.

Solution. Let,

u= In x, and dv= x.dx

du = ½ dx, and v= x2/2

3. Find ∫ x cosX

Solution. Let,

x = u, after differentiation, du/dx = 1

 = x and dv/dx = cosX, already found the value i.e. - du/dx = 1

Now, since dv/dx = cosX

Integrating both sides,

v = ∫ cosx.dx

v = sinX

Formula- ∫u (dv/dx) dx = uv – ∫v (du/dx)dx

∫x cosX.dx = x sinX – ∫ sinX.1 dx 

∫x cosX.dx = x sinX + cosX + c,

where c = constant.

Q: How integration by parts is a useful integration method?

Q: What is the inverse of integration?

Q: What are the types of integration methods?

Q: Name the rules for solving integration by parts problems?

Q: How to remember ∫ u v dx function?