**1.** Try integrating

(1)???

It’s kind of hard, so we try to do something very clever: use the chain rule.

**2.** Recall the chain rule states, if and are differentiable functions, then

(2)

We can apply the fundamental theorem of calculus to both sides

(3)

The left hand side becomes

(4)

But what about the right hand side of Equation (3)?

Let

(5) and

Then the right hand side of (3) becomes

(6)

Observe, we can apply the fundamental theorem of calculus to this monstrosity. We get

(7)

We set this equal to (4), since both Equations (4) and (7) are just manipulations of Equation (3). We find a nice method to perform mildly complicated integrals!

**Example 1.** Recall our motivating problem:

(1)???

Lets let

(8) and

We see we can plug these into Equation (1)

(10)

But we know how to integrate , we see the integral is

(11)

Now we “unroll” our substitution, plugging in which makes (11) become

(12)

then we can evaluate the right hand side on the boundaries as usual.

**Example 2.** Lets consider a more complicated problem:

(12)???

We take

(13) and

Thus we can rewrite (12) as

(14)

This is a trivial integral

(14′)

and we “unroll” the substitution obtaining

(15)

where is a constant of integration.

**Example 3** [Math.SE]**.** Evaluate

(16).

*Solution.* We recall the trigonometric identity

(17)

which makes our integral become

(18).

Now we pick

(19) and .

Our integral (18) becomes

(20).

Now we do the following trick:

(21)

Thus our integral (20) simplifies to

(22).

But these are two easy integrals! We see that

(23)

where is a constant of integration.

The other integral

(24)

requires another substitution, namely,

(25) and

So (24) becomes

(26)

where we simply unroll the substitution.

Now we see that

(27).

*WARNING:* the constant of integration in (27) is *different* than in (23). We just are sloppy/lazy and use the same letter.

But we’re not done yet! We still have to plug in . So we have

(28).

Thus we have

(29).

This concludes our example.

**Exercise 1.** Calculate ???

**Exercise 2** [Math.SE]**.** Calculate

**Exercise 3** [Math.SE]**.** Calculate

**Exercise 4.** Calculate ???

**Exercise 5.** Evaluate .

**Exercise 6.** Compute

**Exercise 7.** Calculate

**Exercise 8.** Consider ???

**Exercise 9** [Math.SE]**.** Evaluate ???

**Exercise 10** [Very Trick!, Math.SE] **.** Compute

**Exercise 11** [Math.SE]**.** Calculate the indefinite integral

**Exercise 12.** Compute . [Hint: multiply by where is some function involving only secant and tangent of .]

**Exercise 13.** Find .

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