Basics of Integration

This should be a refresher from Calculus I, but I'll review what you should know about integrals at this point.


There are, broadly speaking, two big questions in calculus:

  1. How can we describe the way that something is changing?
  2. How can we describe the accumulation of a changing quantity?

The first we handle with the derivative (the main topic of Calculus I), which describes the rate of change of a function at any instant. Graphically, this looks like the slope of the line that is tangent to the function at that instant.

The second question is answered with the integral, the main topic of much of Calculus II. Graphically, this looks like the area beneath the graph of a function ff:

alt text

The Fundamental Theorem of Calculus ties these two together, connecting a function f(x)f(x) and its antiderivative F(x)F(x) (in other words, F(x)=f(x)F'(x)=f(x)): If F(x)=axf(t) dt, then F(x)=f(x) and abf(x) dx=F(b)F(a)\textrm{If } F(x)=\int_a^x f(t)\ dt \textrm{, then } F'(x) = f(x) \textrm{ and } \int_a^b f(x)\ dx=F(b)-F(a)

This is a profound result, tying together the geometric idea of the integral as the area under the curve to the analytical idea of the integral as the antiderivative. This is how we begin to evaluate integrals, by working backwards from known derivatives.

Review Basic Integrals

The Power Rule

In Calculus I, you learned how to differentiate functions like x3x^3: pull the power down in front, and subtract one from the exponent. ddx[x3]=3x2\dfrac{d}{dx} \left[x^3\right] = 3x^2

Therefore, to take the antiderivative of a power function, reverse the process by adding one to the exponent and dividing by that new exponent: x4 dx=15x5+C\int x^4\ dx = \dfrac{1}{5} x^5 + C

In general, this "reverse Power Rule" looks like this: xn dx=1n+1xn+1+C\ans{\int x^n\ dx = \dfrac{1}{n+1} x^{n+1} + C}

Notice that we included +C+C at the end of this answer. That's because, for instance, the derivative of 15x5+C\dfrac{1}{5}x^5 + C will be x4x^4 for any value of CC, so whenever we anti-differentiate, we account for this by adding the arbitrary constant CC.

This reverse power rule works for any power function (including functions that may not initially look like power functions, like x=x1/2\sqrt{x} = x^{1/2} or 1/x3=x31/x^3 = x^{-3}), with one notable exception: the function f(x)=1x.f(x)=\dfrac{1}{x}. Why is this? Try to evaluate this integral using the rule and see what happens: 1x dx=x1 dx=10x0+C\int \dfrac{1}{x}\ dx = \int x^{-1}\ dx = \dfrac{1}{0}x^0 + C

Since this answer includes division by zero, it is not valid. Therefore, we have to handle this case separately. However, this reverse power rule gives us the ability to integrate a broad class of functions (pretty easily, I might add).

The Reciprocal Function

Okay, so let's take care of this reciprocal function. To do so, we simply have to remember that ddx[lnx]=1x,\dfrac{d}{dx} [\ln x] = \dfrac{1}{x},

so when we anti-differentiate the reciprocal function, we get the natural log function: 1x dx=lnx+C\ans{\int \dfrac{1}{x}\ dx = \ln x + C}

The Natural Exponential Function

This is the easiest function to differentiate, and thus the easiest function to integrate:

ddx[ex]=ex, so ex dx=ex+C\dfrac{d}{dx} [e^x] = e^x \textrm{, so } \ans{\int e^x\ dx = e^x + C}

Trigonometric Functions

In Calc I, you just had to bite the bullet and memorize these derivatives, and now you can take advantage of that to reverse the process:

ddx[sinx]=cosxcosx dx=sinx+Cddx[cosx]=sinxsinx dx=cosx+Cddx[tanx]=sec2xsec2x dx=tanx+Cddx[secx]=secxtanxsecxtanx dx=secx+Cddx[cscx]=cscxcotxcscxcotx dx=cscx+Cddx[cotx]=csc2xcsc2x dx=cotx+C\begin{aligned} \dfrac{d}{dx} [\sin x] = \cos x &\longrightarrow \int \cos x \ dx = \sin x + C\ \dfrac{d}{dx} [\cos x] = -\sin x &\longrightarrow \int \sin x \ dx = -\cos x + C\ \dfrac{d}{dx} [\tan x] = \sec^2 x &\longrightarrow \int \sec^2 x \ dx = \tan x + C\ \dfrac{d}{dx} [\sec x] = \sec x \tan x &\longrightarrow \int \sec x \tan x \ dx = \sec x + C\ \dfrac{d}{dx} [\csc x] = -\csc x \cot x &\longrightarrow \int \csc x \cot x \ dx = -\csc x + C\ \dfrac{d}{dx} [\cot x] = -\csc^2 x &\longrightarrow \int \csc^2 x \ dx = -\cot x + C\ \end{aligned}

Basic Rules

Finally, remember a few basic rules: specifically, dealing with constants and addition.

Constants can be pulled out of the integral

Just like with derivatives, constants can be pulled out of an integral, so that what remains can be integrated. In essence, constant multiples just get carried along for the ride, not changing the integral.

cf(x) dx=cf(x) dx\ans{\int c \cdot f(x)\ dx = c \int f(x)\ dx}

ex: 3sinx dx3sinx dx=3cosx+C\textrm{ex: } \int 3 \sin x \ dx 3 \int \sin x \ dx = -3 \cos x + C

(Technically, that answer should be 3cosx+3C-3 \cos x + 3C, but since CC is just an arbitrary constant, 3C3C is also an arbitrary constant, so we can be a bit lazy with that.)

Integrals can be split at addition or subtraction

Again, just as with derivatives, we can split up a function into its terms (pieces that are added/subtracted together) and handle each term separately.

f(x)+g(x) dx=f(x) dx+g(x) dx\ans{\int f(x) + g(x) \ dx = \int f(x) \ dx + \int g(x) \ dx}

ex: x2+2x dx=x2 dx+2x dx=13x3+x2+C\textrm{ex: } \int x^2 + 2x \ dx = \int x^2 \ dx + \int 2x \ dx = \dfrac{1}{3}x^3 + x^2 + C

Summary: Memorize These

Constant Multiple Rule

cf(x) dx=cf(x) dx\int c f(x) \ dx = c \int f(x) \ dx

Addition / Subtraction Rule

f(x)+g(x) dx=f(x) dx+g(x) dx\int f(x) + g(x) \ dx = \int f(x) \ dx + \int g(x) \ dx

Power Rule

xn dx=1n+1xn+1+C\int x^n \ dx = \dfrac{1}{n+1} x^{n+1} + C

Reciprocal Function

1x dx=lnx+C\int \dfrac{1}{x} \ dx = \ln x + C

Natural Exponential Function

ex dx=ex+C\int e^x \ dx = e^x + C

Trig Functions

cosx dx=sinx+Csinx dx=cosx+Csec2x dx=tanx+Csecxtanx dx=secx+Ccscxcotx dx=cscx+Ccsc2x dx=cotx+C\begin{aligned} \int \cos x \ dx &= \sin x + C\ \int \sin x \ dx &= -\cos x + C\ \int \sec^2 x \ dx &= \tan x + C\ \int \sec x \tan x \ dx &= \sec x + C\ \int \csc x \cot x \ dx &= -\csc x + C\ \int \csc^2 x \ dx &= -\cot x + C\ \end{aligned}