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 section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x^2+7x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Differentiate both sides of the equation $u=x^2+7x$
Find the derivative
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 $\frac{e^u\left(2x+7\right)}{2x+7}$ by $2x+7$
Substituting $u$ and $dx$ in the integral and simplify
The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$
Replace $u$ with the value that we assigned to it in the beginning: $x^2+7x$
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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