Chapter 5: Q. 4 (page 428)

State the integration-by-parts formula for definite integrals,
(a) using the notation $d u$ and $d v$ and
(b) without using the notation $d u$ and $d v$

Short Answer

Expert verified

The integration by parts formula for the definite integrals,

Part (a)Using the notation $d u$ and $d v$ is localid="1649212184522" $\int_{a}^{b} u d v = {[u v]}_{a}^{b} - \int_{a}^{b} v d u$

Part (b) Without using the notation $d u$ and $d v$ is $\int_{a}^{b} u (x) v^{ι} (x) d x = {[\begin{matrix} u (x) v (x) \end{matrix}]}_{a}^{b} - \int_{a}^{b} v (x) u^{ι} (x) d x$

Step by step solution

Part (a) Step 1. Given information

Here, we are asked to find the formula for integration by parts for definite integrals using the notations $d u$ and $d v$ .

Part (a) Step 2. Formula using the notation du and dv

If $u$ and $v$ are differentiable functions on $[a, b]$ , then $\int_{a}^{b} u d v = {[u v]}_{a}^{b} - \int_{a}^{b} v d u$

Part (b) Step 1. Given information

Here, we need to state the formula for integration by parts for definite integrals without using the notations $d u$ and $d v$ .

Part (b) Step 2. Formula without using the notation du and dv

If $u (x)$ and $v (x)$ are differentiable functions on $[a, b]$ , then $\int_{a}^{b} u (x) v^{ι} (x) d x = {[u (x) v (x)]}_{a}^{b} - \int_{a}^{b} v (x) u^{ι} (x) d x$

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State the integration-by-parts formula for definite integrals,
(a) using the notation $d u$ and $d v$ and
(b) without using the notation $d u$ and $d v$

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Formula using the notation du and dv

Part (b) Step 1. Given information

Part (b) Step 2. Formula without using the notation du and dv

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State the integration-by-parts formula for definite integrals,(a) using the notationduand dvand (b) without using the notationduanddv

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Formula using the notation du and dv

Part (b) Step 1. Given information

Part (b) Step 2. Formula without using the notation du and dv

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

State the integration-by-parts formula for definite integrals,
(a) using the notation $d u$ and $d v$ and
(b) without using the notation $d u$ and $d v$