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→Fractions: 6 was supposed to be 12, since when changing the expression using Euler’s identity, 1/2 can be factored out from the denominator. The 1/2 is multiplied in the numerator, yielding 2+(e^2ix/2)+4-(e^-2ix). Like terms 4 and 2 can be added to simplify the expression. You could multiply by 2/2, which is why the 1/2 is out of the integral, but the simplfied 6 should be multiplied by 2 as well. Error carry forward for the next question. Tags: Mobile edit Mobile web edit |
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:<math>\int \frac{1+\cos^2 x}{\cos x + \cos 3x} \, dx.</math>
Using Euler's identity, this integral becomes
:<math>\frac12 \int \frac{
If we now make the [[integration by substitution|substitution]] {{math|''u'' {{=}} ''e''<sup>''ix''</sup>}}, the result is the integral of a [[rational function]]:
:<math>-\frac{i}{2}\int \frac{1+
Any [[rational function]] is integrable (using, for example, [[partial fractions in integration|partial fractions]]), and therefore any fraction involving trigonometric functions may be integrated as well.
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