## segunda-feira, 4 de outubro de 2010

### Método de integração

Método de integração: "

From Wikibooks, the open-content textbooks collection Calculus | Integration techniques

 A Wikibookian suggests that Solving Integrals by Trigonometric substitution be merged into this book or chapter. Discuss whether or not this merger should happen on the discussion page.

 ← Integration techniques/Partial Fraction Decomposition Calculus Integration techniques/Tangent Half Angle → Integration techniques/Trigonometric Substitution

If the integrand contains a single factor of one of the forms we can try a trigonometric substitution.

• If the integrand contains let x = asinθ and use the identity 1 − sin2θ = cos2θ.
• If the integrand contains let x = atanθ and use the identity 1 + tan2θ = sec2θ.
• If the integrand contains let x = asecθ and use the identity sec2θ − 1 = tan2θ.

## Contents

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###  Sine substitution

This substitution is easily derived from a triangle, using the Pythagorean Theorem.

If the integrand contains a piece of the form we use the substitution

This will transform the integrand to a trigonometic function. If the new integrand can't be integrated on sight then the tan-half-angle substitution described below will generally transform it into a more tractable algebraic integrand.

Eg, if the integrand is √(1-x2),

If the integrand is √(1+x)/√(1-x), we can rewrite it as

Then we can make the substitution

###  Tangent substitution

This substitution is easily derived from a triangle, using the Pythagorean Theorem.

When the integrand contains a piece of the form we use the substitution

E.g, if the integrand is (x2+a2)-3/2 then on making this substitution we find

If the integral is

then on making this substitution we find

After integrating by parts, and using trigonometric identities, we've ended up with an expression involving the original integral. In cases like this we must now rearrange the equation so that the original integral is on one side only

As we would expect from the integrand, this is approximately z2/2 for large z.

###  Secant substitution

This substitution is easily derived from a triangle, using the Pythagorean Theorem.

If the integrand contains a factor of the form we use the substitution

Find

####  Example 2

Find

We can now integrate by parts