Question

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

Short answer

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

Step-by-step solution

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${\int }_a^{b}udv={\left[\int udv\right]}_a^b={\left[uv\right]}_a^b-{\int }_a^{b}vdu$

Step 2. Explanation

According to theorem 5.10,

${\int }_a^{b}udv={\left[\int udv\right]}_a^b={\left[uv\right]}_a^b-{\int }_a^{b}vdu$

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,${\int }_a^{b}udv={\left[\int udv\right]}_a^b={\left[uv\right]}_a^b-{\int }_a^{b}vdu$