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Sine integral

The formula of the sine integral

\displaystyle \int \sin u \cdot du = - \cos u

Let’s see some examples for integrals of sin x:

 Example 1. Integral of sin 2x

\displaystyle \int \sin(2x) \ dx=

We substitute the 2x for u, we derive and we pass dividing the 2:

u = 2x \quad \Rightarrow \quad du = 2 \ dx \quad \Rightarrow \quad \cfrac{du}{2} = dx

And we replaced the terms of x for u:

\displaystyle \int \sin(u) \cfrac{du}{2}

By properties of integrals we extract the \frac{1}{2} from the integral:

\displaystyle \cfrac{1}{2}\int\sin(u) \ du

Now we apply directly the formula of the sine integral:

\displaystyle \cfrac{1}{2}\int \sin (u) \ du = \left (\cfrac{1}{2}\right)(-\cos(u))

And finally we substitute the u for x and the answer will be:

-\cfrac{1}{2}\cos(2x)

Example 2. Integral of sin2 x

\displaystyle \int \sin^{2}(x) \ dx =

The fastest way to do this integral is to review the formula in the integrals form and you’re done. Another way is the following:

For the resolution of this integral, we need to remember the following trigonometric identity:

\sin^{2}(x) = \cfrac{1}{2} - \cfrac{1}{2} \cos(2x)

Substituting the \sin^{2}(x) for \frac{1}{2}- \frac{1}{2}\cos(2x), we will have the following integral of a sum:

\displaystyle \int \left(\cfrac{1}{2} - \cfrac{1}{2} \cos(2x) \right)dx

By properties of integrals we will have a sum of integrals:

\displaystyle \int \cfrac{1}{2} \ dx + \int - \cfrac{1}{2} \cos(2x) \ dx

The first integral can easily be done, it would be as follows:

\displaystyle \cfrac{1}{2} x + \int - \cfrac{1}{2} \cos(2x) \ dx

We apply properties of the integrals to extract the -\frac{1}{2} from the integral:

\displaystyle \cfrac{1}{2}x - \cfrac{1}{2} \int \cos(2x) \ dx

To solve the second integral, we have to do some procedures that were done in example 1, we substitute 2x for u, we derive and we pass dividing the 2:

u = 2x \quad \Rightarrow \quad du = 2 \ dx \quad \Rightarrow \quad \cfrac{du}{2} = dx

Now we substitute the x for u in the second integral with which we are working:

\displaystyle \cfrac{1}{2}x - \cfrac{1}{2} \int \cos(u) \cfrac{du}{2}

We take the denominator 2 that is in du :

\displaystyle \cfrac{1}{2}x - \cfrac{1}{2}\cfrac{1}{2} \int \cos(u) \ du

We multiply the fractions of \frac{1}{2}:

\displaystyle \cfrac{1}{2} x - \cfrac{1}{4} \int \cos(u) \ du

We apply the cosine integration formula which is the following:

\displaystyle \int \cos(u) \ du =  \sin (u)

Then the integration will remain as follows:

\cfrac{1}{2}x - \cfrac{1}{4} \sin(u)

And finally we substitute the u for 2x to obtain our result:

\cfrac{1}{2}x - \cfrac{1}{4}\sin(2x)

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