Question

How do you evaluate the integral $\int\sqrt{2x + 3}{dx}$ ?

Answer 1

In this question, we need to find the value of $\int\sqrt{2x + 3}{dx}$ . Integration is nothing but its derivative is equal to its original function. Integration is also known as antiderivative. The inverse of differentiation is known as integral. The symbol `$\int$’ is known as the sign of integration. The process of finding the integral of the given function is known as integration. The methods used to integrate the given expression are reverse power rule and reverse chain rule. This method is related to the chain rule of differentiation, which when applied to antiderivatives is known as the reverse chain rule that is integration by u substitution.
Formula Used:
Reverse power rule :
Reverse power rule is used to integrate the expression which is in the form of $x^{n}$
$\int x^{n}{dx} = \dfrac{x^{n + 1}}{n + 1} + c$
Where $c$ is the constant of integration.
This rule is not applicable when $n = - 1$ .
Reverse chain rule :
Reverse chain rule is also known as u-Substitution. U sub is a special method of integration. It is applicable, when the expression contains two functions. This method combines two functions with the help of another variable ‘$u$’ and makes the integration process direct and much easier.
$\int\ f(x)\ f’(x)\ dx$
Here the original component $f(x)$ and the derivative component $f’(x)\ dx$
$\int\ f(x)\ f’(x)\ dx = \int u\ du$
Derivative rule used :
1.$\dfrac{d}{{dx}}\left( kx \right) = k$
2..$\dfrac{d}{{dx}}\left( k \right) = 0$
Where$k$ is a constant.

Complete step-by-step solution:
Given, $\int\sqrt{2x + 3}{dx}$
Let us consider the given function as $I$.
$I = \int\sqrt{2x + 3}{dx}$
Now we need to rewrite $\sqrt{2x + 3}$ in the form of $x^{n}$ .
$\Rightarrow \ I = \int\left( 2x + 3 \right)^{\dfrac{1}{2}}{dx}$ ••• (1)
By using reverse chain rule,
Let us consider $u = 2x + 3$
On differentiating,
We get,
$du = 2dx$
$\Rightarrow \ dx = \dfrac{{du}}{2}$
Thus equation (1) becomes ,
$I = \int\left( u \right)^{\dfrac{1}{2}}\dfrac{\left( {du} \right)}{2}$
$\Rightarrow \ I = \dfrac{1}{2}\int\left( u \right)^{\dfrac{1}{2}}{du}$
Now by using reverse power rule,
We get,
$I = \dfrac{\dfrac{1}{2}(u^{\dfrac{1}{2} + 1})}{\dfrac{1}{2} + 1}+ \ c$
Where $c$ is the constant of integration.
On simplifying,
We get ,
$I =\dfrac{\dfrac{1}{2}{(u)}^{\dfrac{3}{2}}}{\dfrac{3}{2}}+ \ c$
$\Rightarrow \ I = \dfrac{1}{2} \times \dfrac{2}{3}\left( u \right)^{\dfrac{3}{2}}+ \ c$
On further simplifying,
We get,
$I = \dfrac{1}{3}\left( u \right)^{\dfrac{3}{2}}+ \ c$
By substituting the value of $u$,
We get,
$I = \dfrac{1}{3}\left( 2x + 3 \right)^{\dfrac{3}{2}}+ \ c$
Thus we get the value of $\int\sqrt{2x + 3}{dx}$ is equal to $\dfrac{1}{3}\left( 2x + 3 \right)^{\dfrac{3}{2}} + \ c$
Final answer :
The value of $\int\sqrt{2x + 3}{dx}$ is equal to $\dfrac{1}{3}\left( 2x + 3 \right)^{\dfrac{3}{2}} + \ c$.

Note: The concept used in this question is integration method, that is integration by substitution and also with the help of reverse chain rule we can find the integration of the given expression . Since this is an indefinite integral we have to add an arbitrary constant ‘$c$’. $c$ is called the constant of integration. The variable $x$ in $dx$ is known as the variable of integration or integrator. Mathematically, integrals are also used to find many useful quantities such as areas, volumes, displacement, etc. In this question, the derivative rule is also used to solve. Mathematically, a derivative is defined as a rate of change of function with respect to an independent variable given in the function.