Chapter 14, Problem 1RE

To determine

To calculate: The value of integral, ${\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}$.

Answer to Problem 1RE

Solution:

The required value of the integral is $4x^5$.

Explanation of Solution

Given:

The integral is ${\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}$.

Formula used:

The rule of integration:

${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$, where C is integration constant.

Calculation:

Consider the integral ${\displaystyle {\int }_0^{2x}x{y}^{3}dy}$.

Use the below rule of integration:

${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$, where C is integration constant.

Here, the integration is performed with respect to y, so the variable x is treated to be constant.

Now,

$\begin{array}{c}{\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}=x{\displaystyle \underset{0}{\overset{2x}{\int }}{y}^{3}dy}\\ =x{\left[\frac{y^4}{4}\right]}_0^{2x}\\ =\frac{x}{4}\left[{\left(2x\right)}^4\right]\\ =\frac{x}{4}\left[16x^4\right]\end{array}$

Simplify further to get,

${\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}=\frac{x}{4}\left[16x^4\right]$

Hence, the value of the integral is $4x^5$.