I am having problems with this question and I do not know if I am doing it correctly. Can someone assist me?

$$\int_{3}^{4} e^{2x} \,dx $$

$$t = 2x $$

$$dt = 2dx $$

$$\frac {dt}{2} = dx $$

$$\int \frac{1}{2} \cdot e^t + C$$

$$F(b)-F(a) =$$

$$\frac{1}{2}\cdot e^8 – \frac{1}{2}\cdot e^6$$

## Best Answer

I have copied what you wrote to number the steps $$\int_{3}^{4} e^{2x} \,dx \tag{1}$$ $$t = 2x \tag{2}$$ $$dt = 2dx \tag{3}$$ $$\frac {dt}{2} = dx \tag{4}$$ $$\int \frac{1}{2} \cdot e^t + C\tag{5}$$ $$F(b)-F(a) =\tag{6}$$ $$\frac{1}{2}\cdot e^8 - \frac{1}{2}\cdot e^6\tag{7}$$

I agree with steps $(1)-(4)$. However, I don't like the way steps $(5)-(7)$ are written. In particular, it doesn't make sense to include the constant $C$ in step $(5)$ as taking a definite integral never forms an integration constant and you haven't performed the integration at this step. The analysis is step $(6)$ is also incorrect because you couldn't obtain the equality $F(b)-F(a)$ unless you take an antiderivative of an arbitrary function $f$ from the integration limits of $a$ to $b$. However, the final answer in step $(7)$ is correct.

There are two types of integrals (definite and indefinite) encountered in a first course in calculus.

Type 1 (Definite Integrals):These return a number representing the area under the curve $f(x)$ from $x=a$ to $x=b$. By the second fundamental theorem of calculus, we know that if $f$ is continuous on $[a,b]$, and $F$ is an antiderivative of $f$ on $(a,b)$ where $F'(x)=f(x)$, then $$\int_a^bf(x)\,dx=F(b)−F(a).$$ In the above expression, $a$ and $b$ are known as the limits of integration.Type 2 (Indefinite Integrals):These return a function of the independent variable (which is $x$ in your problem). Recall that we may write a general antiderivative as $$\int f(x)\,dx = F(x)+C,$$ where $F$ is the antiderivative of $f$ and $C$ is an integration constant. Observe that we only include the integration constant $C$ provided that we take an indefinite integral without integration limits (otherwise the limits would enforce the condition that we should evaluate the function at a number which will return a number).From these two definitions, it is clear that your integral is a definite integral of the form

$$\int_a^bf(x)\,dx = \int_{3}^{4} e^{2x} \,dx,$$

where $a=3,b=4$, and $f(x)=e^{2x}.$ By the properties of integral calculus, I would write the calculation as

\begin{align*}\int_{3}^{4} e^{2x} \,dx&\stackrel{t=2x}= \frac 12\int_{6}^{8} e^{t} \,dt \\&= \frac 12 \left(e^t\right)\Big|_{t=6}^{t=8}\\&= \frac 12 \left(e^8 - e^6\right). \end{align*}

Also, if we remove the integration limits from your problem, then the integral is now an indefinite integral. The properties of integral calculus tells us that

\begin{align*}\int e^{2x} \,dx&\stackrel{t=2x}=\frac 12\int e^{t} \,dt = \frac 12e^{2x}+C. \end{align*}