The factor ( 1 + x 2) suggests a triangle with base of length 1 and height x : For this triangle, tan θ = x, so we will try the substitution x = tan θ. Then θ = tan − 1 ( x), where we specify − π / 2 < θ <
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Solving Integrals Using Trig Substitution Identify which of the three terms in the table above appear in the integral. Rewrite the equation as necessary to get it into the correct form.
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When using the method of trig substitution, we will always use one of the following three well-known trig identities : (I) 1 − sin 2 θ = cos 2 θ (II) 1 + tan 2 θ = sec 2 θ and (III) sec 2 θ − 1 = tan 2 θ For the expression a 2 − x 2 we use
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Here is a summary for the sine trig substitution. √a2 − b2x2 ⇒ x = a bsinθ, − π 2 ≤ θ ≤ π 2. There is one final case that we need to look at. The next integral will also contain
Look at the triangle in the figure. The radical is the hypotenuse and a is 2, the adjacent side, so Use the results from Steps 2 and 3 to make substitutions in the original
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