# Find Definite Integral $\int_{3}^{4} e^{2x} \,dx$

derivativesintegration

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

$$\int_{3}^{4} e^{2x} \,dx$$
$$t = 2x$$
$$dt = 2dx$$
$$\frac {dt}{2} = dx$$
$$\int \frac{1}{2} \cdot e^t + C$$
$$F(b)-F(a) =$$
$$\frac{1}{2}\cdot e^8 – \frac{1}{2}\cdot e^6$$

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*}