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When do we use the double angle formula in integration?

Question:

When do we use the double angle formula in integration?

Double Angle:

Sometimes in solving equations and even in integrating trigonometric functions it is convenient to transform expressions, such is the case of identities:

{eq}\left\{ \begin{array}{l} se{n^2}(x) = \frac{{1 - \cos (2x)}}{2}\\ {\cos ^2}(x) = \frac{{1 + \cos (2x)}}{2} \end{array} \right. {/eq}

Answer and Explanation:

The double angle formula can be very useful in terms of the integral, one example of this result is the case of the square of sine and cosine:

{eq}\int {se{n^2}(x)\,dx = \int {\,\frac{{1 - \cos (2x)}}{2}\,dx = \frac{1}{2}\left[ {x - \frac{{sen\left( {2x} \right)}}{2}} \right] = \frac{x}{2} - \frac{{sen\left( {2x} \right)}}{4} + C} }\\ \int {{{\cos }^2}(x)\,dx = \int {\,\frac{{1 + \cos (2x)}}{2}\,dx = \frac{1}{2}\left[ {x + \frac{{sen\left( {2x} \right)}}{2}} \right] = \frac{x}{2} + \frac{{sen\left( {2x} \right)}}{4} + C} } {/eq}


Learn more about this topic:

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Indefinite Integral: Definition, Rules & Examples

from Calculus: Tutoring Solution

Chapter 7 / Lesson 14
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