Maths: Integral Calculus: Indefinite Integrals: Methods of integration: Integration by parts Method: Type V : Exercise and Example Solved Problems with Answer, Solution

Integration by parts

Type: V

(1) We know that, if *u* and *v* are two differentiable functions of *x* ,

Integrate on both sides with respect to *x* .

This method is very useful when the integrand is a product of two different types of functions or a function which is not directly integrable. The success of this method depends on the proper choice of *u* . So we can choose the function *u* by using the following guidelines.

(i) If the integrand contains only a function which is directly not integrable, then take this as *u*.

(ii) If the integrand contains both directly integrable and non integrable functions, then take non integrable function as *u*.

(iii) If the integrand contains both the functions are integrable and one of them is of the form *x* *n* ,*n* is a positive integer, then take this *xn* as *u*.

(iv) for all other cases, the choice of *u* is ours.

(Or) we can also choose *u* as the function which comes first in the word “I L A T E” Where,

I stands for the inverse trigonometric function

L stands for the logarithmic function

A stands for the algebraic function

T stands for the trigonometric function

E stands for the exponential function and

take the remaining part of the function and *dx* as *dv*.

(2) If *u* and *v* are functions of *x* , then ∫*udv* = *uv* − *u*′*v*1 + *u*′′*v* 2 − *u*′′′*v*3 + ...

where *u* ′ , *u*′′, *u*′′′, . . . are the successive derivatives of *u* and *v*1* *, *v*2* *,* v*3* *, . . .* *are the repeated integrals of* v*.

Note

The above mentioned formula is well known as Bernoulli’s formula.

Bernoulli’s formula is applied when *u* = *xn* where *n* is a positive integer.

Example

Exercise 2.5

Integrate the following with respect to *x.*

1. xe−x

2. x3 e3x

3. log x

4. x log x

5. xn log x

6. x5ex2

Answer:

Tags : Exercise and Example Solved Problems with Answer, Solution | Methods of integration , 12th Business Maths and Statistics : Chapter 2 : Integral Calculus - I

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12th Business Maths and Statistics : Chapter 2 : Integral Calculus - I : Integration by parts Method | Exercise and Example Solved Problems with Answer, Solution | Methods of integration

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