Indefinite Integrals Exercises

Answer

(a) The most complicated "inside" function is 6x + 4, so let

u = 6x + 4

u' = 6

(b) First we change variables. We have to break up 30 into 5 ⋅ 6 so we can find u':

We use an appropriate pattern to integrate:

And finally put the original variable back in:

(u)⁵ + C = (6x + 4)⁵ + C

We conclude

Answer

(a) The exponent of 4 is the most complicated inside function:

so we have

u = x² + 4u' = 2x

(b) We change variables, integrate, and put the original variable back in:

Answer

(a) The most complicated "inside" function is the e^x inside the sine function:

The derivative of e^x is e^x, so

u = e^xu' = e^x

(b) Change variables, integrate, and change variables back:

Answer

The most complicated function that's still inside something is the denominator of the fraction, which is "inside" the operation :

Let u = x + 1. Then u' = 1. We have

Answer

The most complicated inside function is the exponent of e:

Let u = sin x. Then u' = cos x.

Answer

This integral doesn't require substitution. It only requires thinking backwards.

What has a derivative of -cos x?

The answer is -sin x. So 

Answer

We can rewrite this integral as

This makes it more obvious that we should choose u = sin x and u' = cos x.

Answer

The most complicated inside function is the exponent of 3, so let u = (2x² + 4x) and then u' = (4x + x).

Answer

Let u = (2x + 3). Then u' should be 2. In order to see 2 as a factor of the integrand, we have to break up the coefficient 20:

Now we have to think backwards. What has a derivative of 10u⁴? Answer: 2u⁵. So

Answer

This doesn't require substitution. Simplify the integrand by raising 5 and x to the 4th power:

What has a derivative of 625x⁴? Answer:

So

Answer

If we only looked at the first guideline for how to choose u, we would take cos (ln x) as u since that's the most complicated thing that's still "inside" something else (it's the numerator of the fraction). However, the derivative of cos (ln x) is not a factor of the integrand, so that won't work. Let's try making u a little less complicated. Let

Then

There. That worked.

Answer

 Rewrite the integral:

We have two choices. If we let u = sin x and u' = cos x, we get the integral

which isn't very helpful. On the other hand, if we let u = cos x, then u' = -sin x and we get an integral we know what to do with:

Answer

The most obvious choice of "inside" function is u = (3x² + 4x + 7).

Then u' = (6x + 4).

Answer

Let

u = 3x + 4

u' = 3

We multiply by 3 inside the integral and  outside of it. Since  we're not changing the value of the expression:

We integrate, writing C instead of , and put the original variable back in:

Answer

Let

In this case we need the factor  inside the integral, and its reciprocal 4 outside the integral.

Answer

Let

u = e^4x

u' = 4e^4x

We need to introduce the factor 4 to the integrand, so we multiply the integrand by 4 and the outside of the integral by .

Those parentheses in the second-to-last step are important. If we had written

instead of

we would have gotten the wrong answer.

Answer

Let u = (3x² + 4) and u' = 6x. Then we need to introduce 6 to the integrand and  to the outside of the integral.

Answer

Let so .

Unfortunately the constant in our integrand is , but we can factor it like so:

.

Now we're just left with another which we can pull out of the integral:

.

And the resulting integral is

.

It's time to get that u out of there, though. After substituting everything back in, we can say

.

Answer

Let . Since , we have

We rewrite the integrand:

This is almost what we want, but the sign is wrong. Since (-1)(-1) = 1, let's multiply the integrand by (-1)(-1). Then we can see u' in the integrand:

Answer

Let u = x² + 4x and u' = 2x + 4. If we multiply the numerator of the integrand by 2, we get u'.

Answer

Following the hint, let's rewrite the integrand:

We choose u to be the denominator, because we know how to find the integral of , but we don't know how to find the integral of . So let

u = sin(4x)

u' = 4cos(4x).

The integrand is missing a factor of 4, and we know how to fix that.