Q. 4 - Calculus

Question

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

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

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

Step-by-Step Solution

Answer

The integration by parts formula for the definite integrals,

Part (a) Using the notation $du$ and $dv$ is ${\int }_a^{b}udv = {\left[uv\right]}_a^b - {\int }_a^{b}vdu$

Part (b) Without using the notation $du$ and $dv$ is ${\int }_a^{b}u\left(x\right)v^{\iota }\left(x\right)dx = {\left[\begin{array}{c}u\left(x\right) v\left(x\right)\end{array}\right]}_a^b -{\int }_a^{b}v\left(x\right) u^{\iota }\left(x\right)dx$

Part (a) Step 1. Given information

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

Part (a) Step 2. Formula using the notation d u and d v

If $u$ and $v$ are differentiable functions on $\left[a,b\right]$, then ${\int }_a^{b}udv = {\left[uv\right]}_a^b - {\int }_a^{b}vdu$

Part (b) Step 1. Given information

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

Part (b) Step 2. Formula without using the notation d u and d v

If $u\left(x\right)$ and $v\left(x\right)$ are differentiable functions on $\left[a,b\right]$, then ${\int }_a^{b}u\left(x\right) v^{\iota }\left(x\right)dx = {\left[u\left(x\right) v\left(x\right)\right]}_a^b - {\int }_a^{b}v\left(x\right) u^{\iota }\left(x\right)dx$