Question

How do you use substitution to integrate $(2x{({x^2} + 1)^{23}})dx$ ?

Answer 1

Hint: Integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions in smaller parts. Differentiation is the process of finding the derivative and integration is the process of finding the antiderivative of a function. So, these processes are inverse of each other. In this question we integrate the function by substitution in this we use the concepts of both integration and differentiation.

Complete step by step answer:
In this method of integration by substitution, any given function is transformed into a simple form of integral by substituting the independent variable by others and this method is extremely useful when we make a substitution for a function whose derivative is also present in the integrand. Integrand is used for the term which is to be integrated. We have the function $(2x{({x^2} + 1)^{23}})dx$ which is to be integrated by the method of substitution.Now, $\int {(2x({x^2}} + 1{)^{23}})dx$. By using the substitution method. Let,
$t = ({x^2} + 1)$
We have taken $t = ({x^2} + 1)$ as we have its derivative $2x$ in the given function.

Now we will do differentiation of the term which is to be substituted. We get,
$dt = 2xdx$
So, we can write the original integral in terms of $x$ into a new equivalent integral in terms of $t$ as follows
$\int {{t^{23}}} dt$
We know that $\int {{u^n}du} = \dfrac{{{u^{n + 1}}}}{{n + 1}} + C$
So,
$ \Rightarrow \int {{t^{23}}} dt = \dfrac{{{t^{23 + 1}}}}{{23 + 1}} + C$
$ \Rightarrow \int {{t^{23}}} dt = \dfrac{{{t^{24}}}}{{24}} + C$
After putting value of $t$ in terms of $x$, we obtain the final answer as $\dfrac{{{{({x^2} + 1)}^{24}}}}{{24}} + C$.

Hence, the integration of $(2x{({x^2} + 1)^{23}})dx$ is $\dfrac{{{t^{24}}}}{{24}} + C$.

Note: The integration is the process of finding the antiderivative of a function. It is a similar way to add the slices to make it whole. The integration is the inverse process of differentiation. Integration is also called the anti-differentiation. The integration is used to find the volume, area and the central values of many things. Integration by substitution method is also known as “ Reverse Chain Rule”. This method is extremely useful when we make a substitution for a function whose derivative is also included in the integer. With this, the function simplifies and then the basic integration formula can be used to integrate the function.