Definite integrals vs. indefinite integrals

Recall that 1st FTC says if you find a function $F(x)$ such that $F'(x) = f(x),$ then we can compute $$\int_{a}^{b}f(x)dx = F(b) - F(a).$$ For example, when $f(x) = \sin(x),$ we may find $F(x) = -\cos(x)$ to compute $$\int_{0}^{\pi}\sin(x)dx = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2.$$ However, what if we use $F(x) = -\cos(x) + 3000$ instead? We still have $$F'(x) = -(-\sin(x)) + 0 = \sin(x) = f(x),$$ so does 1st FTC work for this $F(x)$ as well? In fact, it does work. If we apply 1st FTC with this $F(x) = -\cos(x) + 3000,$ we have $$\begin{align*}\int_{0}^{\pi}\sin(x)dx &= -\cos(\pi) + 3000 - (-\cos(0) + 3000) \\ &= -(-1) + 3000 - (-1 + 3000) \\ &= 1 - 3000 + 1 + 3000 \\ &= 2.\end{align*}$$ Aha! That $3000$ cancelled out with another $3000.$ Notice that we could use $F(x) = -\cos(x) + c$ for any constant $c.$ In fact, this is the full list of the functions $F(x)$ such that $F'(x) = -\cos(x).$

Tip. When you find a single $F(x)$ with $F'(x) = f(x),$ then the full list of the functions whose derivative is $f(x)$ is given by $F(x) + c$ with all constants $c.$

Exercise (cf. p.322). What are the functions whose derivatives are $0$? (Hint: intuitively, these are functions with no variation. Such functions should be called constant.)

Takeaway. The derivative determines a function up to a change in constant (or geometrically, change in height or vertical shift).

We denote by $$\int \sin(x) dx$$ the list of all antiderivates of $\sin(x).$ We have seen that this list can be given as $$\int \sin(x) dx = -\cos(x) + c$$ where $c$ is any fixed constant. This list is called the indefinite integral of $\sin(x).$ Note that this is NOT a definite integral. That is, when we are given two other values $a$ and $b,$ then $$\int_{a}^{b}\sin(x)dx = -\cos(b) + \cos(a),$$ which is a real number.

Exercise. Compute $$\int x^{3000} dx.$$

Exercise. Compute $$\int_{-3000}^{3000} x^{321} dx.$$

Exercise (cf. p.323). There are two things wrong about the following formula where $n$ is any integer: $$\int x^{n} dx = \frac{x^{n+1}}{n+1}.$$ What are they?

Hard exercise. There is one thing wrong about the following formula: $$\int \frac{1}{x} dx = \ln(x) + c,$$ where $c$ is any fixed constant. What is it? (Hint: if you get stuck, read p.324. Make sure you understand the answer. Please consult me if it is too difficult.)

Exercise. Compute $$\int_{1}^{e}\frac{1}{x}dx.$$

Exercise. Compute $$\int_{-e}^{-1}\frac{1}{x}dx.$$

Very bad exercise. Compute $$\int \left(\int \cos(x) dx\right) dx.$$ (Note: this is NOT what people call a "double integral", which is dealt in Math 215. Please just take this as a joke and skip this problem. The correct answer is will be given by $\int \sin(x) + c dx,$ where $c$ can be any fixed constant.)

Exercise. Compute $$\int_{-1}^{1}2x + 3x^{2} + 4x^{3} + 5x^{4}dx.$$ (Hint: recall odd and even functions.)

Exercise. Compute $$\int 2x + 3x^{2} + 4x^{3} + 5x^{4}dx.$$

Exercise. Compute the derivative $$\frac{d}{dx} \tan(x).$$ Then read Example 4 on p.326.