Page 114 - Calculus Demystified

CHAPTER 4
and The Integral 101
e 2x
e 2x dx = + C.
2

The symbol is called an integral sign (the symbol is in fact an elongated “S”)
and the symbol “dx” plays a traditional role to remind us what the variable is. We
call an expression like

f(x) dx

an indefinite integral. The name comes from the fact that later on we will have a
notion of “definite integral” that specifies what value C will take—so it is more
definite in the answer that it gives.

EXAMPLE 4.1
Calculate

sin(3x + 1)dx.

SOLUTION
We know that we must guess a trigonometric function. Running through
the choices, cosine seems like the best candidate. The derivative of cos x is
− sin x. So we immediately see that − cos x is a better guess—its derivative
is sin x. But then we adjust our guess to F(x) =− cos(3x + 1) to take into
account the form of the argument. This almost works: we may calculate that

F (x) = 3 sin(3x + 1). We determine that we must adjust by a factor of 1/3.
Now we can record our final answer as

1
sin(3x + 1)dx =− cos(3x + 1) + C.
3
We invite the reader to verify that the derivative of the answer on the right-hand
side gives sin(3x + 1).
EXAMPLE 4.2
Calculate

x
dx.
2
x + 3
SOLUTION
We notice that the numerator of the fraction is nearly the derivative of the
denominator. Put in other words, if we were asked to integrate
2x
2
x + 3