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CALCULUS 1 PROBLEMS & SOLUTIONS

6.4
The Indefinite Integral


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Review


1. The Indefinite Integral

The fundamental theorem of calculus (see Section 6.3 Theorem 1) provides us a powerful method to

squarely on finding an antiderivative F of f.


Definition 1
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The general antiderivative of a function f is also called the indefinite integral of f, and is denoted:



Note that there are no limits of integration in that notation. Antidifferentiation is also called indefinite
integration, or simply integration.
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2. Integration By Inspection

The general antiderivative of f(x) = 2x is obviously F(x) = x² + C, where C is a constant. We obtain
F by inspection: the derivative of x² is 2x; so we insert C to it to obtain F. We can insert C because
the derivative of any constant is always 0. We must insert C because we're dealing with the general
antiderivative. So, ò 2x dx = x² + C. Integration by inspection requires that we know differentiation
formulas. We inspect to see which function has the derivative as 2x. As another example, we have
ò cos x dx = sin x + C, because we know that (d/dx) sin x = cos x; thus we insert C to sin x to get
ò cos x dx. Remember to insert the constant C. One more example:

where we use a trigonometric identity to render the integrand obvious for inspection. Part 3 below
gives a table of integration formulas corresponding to some differentiation formulas.

Now, the integral ò (x/(x² + 1)) dx cannot be evaluated by inspection. There's a special technique to
find it. The next several sections present various techniques to find integrals that cannot be evaluated
by inspection. In this section, we handle only integrals that can be evaluated by inspection.

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3. A Table Of Some Integrals





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Problems & Solutions


1. Evaluate the following indefinite integrals.


Solution



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2. Evaluate the indefinite integral ò sin⁴ x dx.

Solution



Note

We insert the constant C only after the last integral has been evaluated. There's no point to insert it
before then, because if we did, then we should add up those constants to get just a single constant.

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