integration analysis

Consider the integral

₀² (x² + 3x)/(x + 2) dx

This is a definite integral, so we need to find the area under the integrand f(x), which we do by using the Fundamental Theorem of Calculus: we find an antiderivative, evaluate it at the endpoints of the integral, and take the difference of the values.

To find an antiderivative of f(x), we go through our list of integration methods:

Recognize elementary antiderivatives
Rewrite the integrand to make it easier
Use substitution to reverse the chain rule or simplify the integrand
Use integration by parts
Use inspection to see the value of a definite integral

to find one that works for this integrand. In this case, only one method is appropriate. We can rewrite the integrand: use long division to divide the denominator into the numerator to give an expression that is more easily integrable. This is shown below.

Rewriting:
Use long division to divide the denominator into the numerator to rewrite the integral:

(x² + 3x)/(x + 2) = x + 1 - 2/(x + 2).

This can be broken into separate integrals,

(x² + 3x)/(x + 2) dx =

xdx+

1dx+

-2/(x + 2)dx

which can be evaluated using basic antidifferentiation, basic antidifferentiation, and substitution, respectively, to obtain

xdx+

1dx+

-2/(x + 2)dx = ((1/2) x²) + (x) + (-2 ln(x + 2)) + C.

Explanation for rewritten term(s)
Basic antidifferentiation
In this case, by using rules for basic antiderivatives we can just write down the antiderivative of the integrand:

x dx = (1/2) x² + C

Basic antidifferentiation
In this case, by using rules for basic antiderivatives we can just write down the antiderivative of the integrand:

1 dx = x + C

Substitution
Let w = x + 2. Then w' = 1, so dw = dx. The integral can therefore be rewritten as

-2/(x + 2) dx =

-2/w dw = -2 ln(w) + C

Thus, substituting back for w,

-2/(x + 2) dx = -2 ln(x + 2) + C

To evaluate the definite integral, we take this antiderivative, evaluate it at the endpoints of the integral (0 and 2), and take the difference of the values. This gives

[ ((1/2) (2)²) + ((2)) + (-2 ln((2) + 2)) ] - [ ((1/2) (0)²) + ((0)) + (-2 ln((0) + 2)) ] = 4 - ln(4).

integration tutorial: analysis