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# The Indefinite Integral Help

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## Introduction to The Indefinite Integral

In practice, it is useful to have a compact notation for the antiderivative. What we do, instead of saying that “the antiderivative of f (x) is F(x) + C ,” is to write

f (x) dx = F(x) + C.

So, for example,

∫ cos x dx = sin x + C

and

and

The symbol f 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 1

Calculate

∫ sin(3x + 1)dx .

#### Solution 1

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

We invite the reader to verify that the derivative of the answer on the right-hand side gives sin(3x + 1).

Calculate

#### Solution 2

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

then we would see that we are integrating an expression of the form

(which we in fact encountered among our differentiation rules in Section 2.5). As we know, expressions like this arise from differentiating log (x).

Returning to the original problem, we pose our initial guess as log[x2 + 3]. Differentiation of this expression gives the answer 2x /[ x2 + 3]. This is close to what we want, but we must adjust by a factor of 1/2. We write our final answer as

You Try It: Calculate the indefinite integral

#### Example 3

Calculate the indefinite integral

#### Solution 3

We observe that the expression 6x2 + 4x is nearly the derivative of x3 + x2 + 1. In fact if we set (x) = x3 + x2 + 1 then the integrand (the quantity that we are asked to integrate) is

[ (x)]50 · 2 ′ (x).

It is natural to guess as our antiderivative [ (x)]5. Checking our work, we find that

([ (x)]51)′ = 51[ (x)]50 · ′ (x).

We see that the answer obtained is quite close to the answer we seek; it is off by a numerical factor of 2/51. With this knowledge, we write our final answer as

You Try It: Calculate the indefinite integral

Find practice problems and solutions for these concepts at: The Integral Practice Test.

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