Q. 3 - Calculus

Question

State the integration-by-parts formula for indefinite integrals,

(a) using the notation $du$ and $dv$ and

(b) without using the notation $du$ and $dv$.

Step-by-Step Solution

Answer

The formula of integration by parts for indefinite integrals

Part (a) Using the notation $du$ and $dv$ is $\int udv=uv-\int vdu$

Part (b) Without using the notation $du$ and $dv$ is $\int u\left(x\right)v^{\iota }\left(x\right) dx = u\left(x\right)v\left(x\right)-\int v\left(x\right)u^{\iota }\left(x\right)dx$

Part (a) Step 1. Given information

• We have to state the formula of integration by parts for the indefinite integrals using $du$ and $dv$.
• Indefinite integrals are those integrals which do not have any upper limit and lower limit.

Part (a) Step 2. Formula with d u and d v

The formula of integration by parts for indefinite integrals containing the notations $du$ and $dv$ is $\int udv=uv-\int vdu$

Part (b) Step 1. Given Information

• We have to state the formula of integration by parts for the indefinite integrals using $du$ and $dv$.
• Indefinite integrals are those integrals that do not have any upper limit and lower limit.

Part (b) Step 2. Formula with the notations d u and d v

The formula of integration by parts for indefinite integrals not containing the notations  $du$ and $dv$ is, $\int u\left(x\right)v^{\iota }\left(x\right) dx = u\left(x\right)v\left(x\right)-\int v\left(x\right)u^{\iota }\left(x\right)dx$