Solved example of integrals of exponential functions
We can solve the integral $\int e^{\left(x^2+7x\right)}\left(2x+7\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a part of the integral with a new variable, which when substituted makes the integral easier. We see that $e^{\left(x^2+7x\right)}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Find the derivative
Applying the derivative of the exponential function
The derivative of a sum of two functions is the sum of the derivatives of each function
The derivative of the linear function times a constant, is equal to the constant
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Simplify the fraction by $u\left(2x+7\right)$
Substituting $u$ and $dx$ in the integral and simplify
The integral of a constant is equal to the constant times the integral's variable
Substitute $u$ back with the value that we assigned to it: $e^{\left(x^2+7x\right)}$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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