Indefinite Integrals Exercises

Answer

Answer. We want to choose u so that u' is simpler than u. This is probably a good choice:

u = x

v' = sin x

Now u' = 1 is simpler than u and v = (-cos x) isn't any worse than v'.

We apply the formula:

Simplifying the right-hand side and evaluating the new integral gives us

We could also write this as

It's possible that after applying the formula for integration by parts, you'll need to use integration by parts again in order to figure out the new integral. If that happens, be very careful with signs and coefficients. We recommend you work out the new integral by itself, then wrap up your expression for the new integral in parentheses before putting it back into the formula.

Answer

Take

so that u' is simpler than u and v isn't any worse than v':

u' = x

v = sin x

Sticking everything in the formula, we get

Using integration by parts again, we find that

Wrap this up in parentheses, and put it back in the integration by parts formula where we left off:

We don't care about including the + C in the parentheses, since it doesn't matter if we end up with + C or – C.

If we hadn't done that, it would have been very easy to write

which is not the correct answer (the term x cos x has the wrong sign). This is the step we're talking about when we say "be careful with your signs and coefficients" and "wrap the expression for the new integral in parentheses before putting it back in the formula."

Answer

We want u' to be simpler than u, so choose

u = x²

v' = sin x

Then

u' = 2x

v = -cos x

We use the formula for integration by parts:

Now we need to go figure out

This requires integration by parts again.

Take

u = x

v' = cos x

Then

u' = 1

v = sin x

Applying the integration by parts formula,

Now that we know

we can go pick up where we left off. We don't bother including the + C in the parentheses, since having + C instead of + 2C doesn't change the final answer.