7.1 Integration by Parts

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7.1 Integration by Parts

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Integration by Parts

Every differentiation rule has a corresponding integration rule. For instance, the Substitution Rule for integration

corresponds to the Chain Rule for differentiation. The rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts.

The Product Rule states that if f and g are differentiable functions, then

[f(x)g(x)] = f(x)g′(x) + g(x)f′(x)

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Integration by Parts

In the notation for indefinite integrals this equation becomes

∫

^[f(x)g′(x) + g(x)f′(x)] dx = f(x)g(x) or

∫

^f(x)g′(x) dx +

∫

^g(x)f′(x) dx = f(x)g(x) We can rearrange this equation as

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Integration by Parts

It is perhaps easier to remember in the following notation.

Let u = f(x) and v = g(x). Then the differentials are du = f′(x)dx and dv = g′(x)dx, so, by the Substitution Rule, the formula for integration by parts becomes

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Example 1

Find

∫

x sin x dx.

Solution Using Formula 1:

Suppose we choose f(x) = x and g′(x) = sin x. Then f′(x) = 1 and g(x) = –cos x. (For g we can choose any antiderivative of g ′.) Thus, using Formula 1, we have

∫

x sin x dx = f(x)g(x) –

∫

g(x)f′(x) dx

= x(–cos x) –

∫

(–cos x) dx

= –x cos x +

∫

cos x dx

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Example 1 – Solution

It’s wise to check the answer by differentiating it. If we do so, we get x sin x, as expected.

Solution Using Formula 2:

Let

u = x dv = sin x dx

Then du = dx v = –cos x

cont’d

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Example 1 – Solution

So

∫

x sin x dx =

∫

^{x sin x dx}

= x (–cos x) –

∫

(–cos x) dx

= –x cos x +

∫

^{cos x dx}

= –x cos x + sin x + C

u v u du

cont’d

u dv

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Integration by Parts

If we combine the formula for integration by parts with Part 2 of the Fundamental Theorem of Calculus, we can evaluate definite integrals by parts.

Evaluating both sides of Formula 1 between a and b, assuming f′ and g′ are continuous, and using the

Fundamental Theorem, we obtain