Q 14. - Calculus

Question

Provide a justification for each equality in the statement of the integration-by-parts formula for definite integrals from Theorem 5.10.

Step-by-Step Solution

Answer

The definite integral of is a number and represents the area under the curve from a to b.

Step 1. Given information

Theorem 5.10 is as follows:

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

Step 2. Explanation

According to theorem 5.10,

${\int }_a^{b}u dv={\left[\int u dv\right]}_a^b={\left[uv\right]}_a^b-{\int }_a^{b}v du$

Here, $u\left(x\right),v\left(x\right)$ are differentiable functions on $\left[a,b\right]$

The definite integral of is a number and represents the area under the curve from a to b.

Take $u,dv$ as first and second functions respectively.

So, ${\int }_a^{b}u dv={\left[\int u dv\right]}_a^b={\left[uv\right]}_a^b-{\int }_a^{b}v du$