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Studying for a test? Prepare with these 9 lessons on Antiderivatives and the fundamental theorem of calculus.

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# Indefinite integrals of sin(x), cos(x), and eˣ

Video transcript

I thought I would do
a few more examples of taking antiderivatives, just
so we feel comfortable taking antiderivatives of all
of the basic functions that we know how to
take the derivatives of. And on top of that, I
just want to make it clear that it doesn't always
have to be functions of x. Here we have a function
of t, and we're taking the antiderivative
with respect to t. And so you would
not write a dx here. That is not the notation. You'll see why when we
focus on definite integrals. So what's the antiderivative of
this business right over here? Well, it's going to be the same
thing as the antiderivative of sine of t, or the indefinite
integral of sine of t, plus the indefinite integral,
or the antiderivative, of cosine of t. So let's think about what
these antiderivatives are. And we already know a
little bit about taking the derivatives
of trig functions. We know that the derivative
with respect to t of cosine of t is equal to negative sine of t. So if we want a
sine of t here, we would just have to take the
derivative of negative cosine t. If we take the derivative
of negative cosine t, then we get positive sine of t. The derivative with
respect to t of cosine t is negative sine of t. We have the negative out front. It becomes positive sine of t. So the antiderivative of sine
of t is negative cosine of t. So this is going to be equal
to negative cosine of t. And then what's the
antiderivative of cosine of t? Well, we already know that
the derivative with respect to t of sine of t is
equal to cosine of t. So cosine of t's
antiderivative is just sine of t-- so plus sine of t. And we're done. We've found the antiderivative. Now let's tackle this. Now we don't have a t. We're taking the indefinite
integral with respect to-- actually,
this is a mistake. This should be
with respect to a. Let me clean this up. This should be a da. If we were taking this
with respect to t, then we would treat all of
these things as just constants. But I don't want to
confuse you right now. Let me make it clear. This is going to be da. That's what we are
integrating or taking the antiderivative
with respect to. So what is this
going to be equal to? Well once again, we can rewrite
it as the sum of integrals. This is the indefinite
integral of e to the a da, so this one right
over here-- a d I'll do it in green-- plus
the indefinite integral, or the antiderivative,
of 1/a da. Now, what is the
antiderivative of e to the a? Well, we already know a
little bit about exponentials. The derivative with
respect to x of e to the x is equal to e to the x. That's one of the reasons why
e in the exponential function in general is so amazing. And if we just replaced
a with x or x with a, you get the derivative with
respect to a of e to the a is equal to e to the a. So the antiderivative
here, the derivative of e to the a, the antiderivative
is going to be e to the a. And maybe you can shift it
by some type of a constant. Oh, and let me
not forget, I have to put my constant
right over here. I could have a constant factor. So let me-- always important. Remember the constant. So you have a constant
factor right over here. Never forget that. I almost did. So once again, over here,
what's the antiderivative of e to the a? It is e to the a. What's the
antiderivative of 1/a? Well, we've seen that
in the last video. It is going to be the natural
log of the absolute value of a. And then we want to have the
most general antiderivative, so there could be a constant
factor out here as well. And we are done. We found the antiderivative
of both of these expressions.