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What is the integral of e7x?

Answer
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Hint: From the question given, we have been asked to find the integral of e7x. To solve this question, we have to know the basic concepts of integration. We have to use the substitution method to solve the given question. So, we will take u=7x and then proceed.

Complete step by step answer:
Let’s learn what integration is before understanding the concept of integration by substitution. The integration of a function f(x) is given by F(x) and it is represented by:
f(x)dx = F(x) + C
Substitution method: In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable by others. This method is also known as u- substitution or change of variables, is a method for evaluating integrals and antiderivatives.
Here in this case, e7x
We are going to use u- substitution.
Let u = 7x
Differentiate(derivative) both parts:
du=7dx
du7=dx
Now we can replace everything in the integral:
17eudu
Bring the constant upfront
17eudu
The integral of eu is simply eu
17eu
And replace the u back
17e7x
There is also a shortcut you can use:
Whenever you have a function of which you know the integralf(x), but it has a different argument
the function is in the form f(ax±b)
If you want to integrate this, it is always equal to 1aF(ax+b), where F is the integral of the regular f(x) function.
In this case:
f(x)=ex
F(x)=exdx=ex
a=7b=0f(ax+b)=e7x
e7xdx=1aF(ax+b)=17e7x

Note: Students should be well known about the concept of integration. Students should know the formulas in integration. Students should know the method of substitution. Students should be careful while performing substitution methods. Students should be careful while calculating the problem.