Integral of sin(x)

In this post we saw the most important integrals to memorize in order to solve more complex ones. One of them is the integral of sin(x), which is -cos(x). But why? In a previous post we saw why the antiderivative of cos(x) is sin(x), and we are going to do the same here.

f(x)=sin(x) (in red) and its antiderivative, F(x)=-cos(x) (in blue).

Definition of sin(x)

As we saw at the end of the previous post, we can define sin(x) using exponential functions and, in particular, Euler’s identity:

e is Euler’s number and i is the imaginary unit.; e^(ix) is Euler’s identity:

We can prove that sin(x) can be expressed as shown above:

Now that we know that the two expressions are equivalent, we can confirm that

and we know how to evaluate the second integral:

Linearity and constant rule are applied and now we have two integrals, very easy to solve using u-substitution. You actually don’t need to perform u-sub: just rewrite the function and divide it by the exponent without the x:

Don’t forget to add the +C at the end. I didn’t write it to isolate the result. If you really want to use u-sub, simply let u=ix for the first integral and u=-ix for the second; you will see that it’s the exact same thing.

Anyway, that’s the result. cos(x) can be written as

This is the expression we got from the integral, except it’s got the minus sign, so

This is a demostration why the integral of sin(x) is cos(x).

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