Calculus 2, Fall 2016

## Prelude to substitution

1. Does every elementary function have an elementary antiderivative?

2. Indefinite integrals are _______________, so to evaluate indefinite integrals we need to be able to undo ________________.

3. Is it possible to use graph transformations to simplify area problems?

4. Is it possible to use a change of variables to solve integrals?

5. Is it possible to undo the chain rule for derivatives and simplify evaluating all integrals?

## An area problem

Simplify the area problem by finding another parabola that has the same area under it as the given parabola. Draw it!

## Another area problem

Find the area between the function $$f(t) = (2t-1)^2$$ and the $$t$$-axis for $$0 \leq t \leq 3$$. Graph the function and then simplify the area problem by finding another parabola that has the same area under it as the given parabola.

## Another area problem

These graphs show that for the change of coordinates $$u = 2t-1,$$ $\int_{t=0}^{t=3} (2t-1)^2 \, dt = \int_{u=-1}^{u=5} \frac{u^2}{2} \, du.$

## Substitution

Let $$F$$ and $$g$$ be differentiable functions where the range of $$g$$ is an interval contained in the domain of $$F$$. If we write $$u = g(x)$$, then $\begin{array}{rcl} \int F'(g(x)) g'(x) \, dx & = & \int F(u) \, du \\ & = & F(u) + C \\ & = & F(g(x)) + C \end{array}$ and $\begin{array}{rcl} \int_{x=a}^{x=b} F'(g(x)) g'(x) \, dx & = & \int_{u=g(a)}^{u=g(b)} F'(u) \, du \\ & = & F(g(b)) - F(g(a)). \end{array}$

## Basic examples

1. $$\displaystyle \int 3x^2 \cos\left(x^3\right) \, dx$$

2. $$\displaystyle \int t^3 \sqrt{t^4 + 5} \, dt$$

3. $$\displaystyle \int 3 (e^{2x} - 4x^7) \, dx$$

4. $$\displaystyle \int \frac{y^4}{y^5+1} \, dy$$

5. $$\displaystyle \int \frac{1}{9+x^2} \, dx$$

## More challenging examples

1. $$\displaystyle \int_0^4 w \sqrt{4-w} \, dw$$

2. $$\displaystyle \int \frac{x+1}{x^2+2x+19} \, dx$$

3. $$\displaystyle \int \frac{ e^{\sqrt{x}} }{ \sqrt{x} } \, dx$$

4. $$\displaystyle \int \sqrt{1 + \sqrt{x}} \, dx$$

5. $$\displaystyle \int \frac{x + 2 \sqrt{x-1}}{ 2x \sqrt{x-1}} \, dx$$

## Definite integrals

1. $$\int_0^{\pi/4} \tan(\theta) \, d\theta$$

2. $$\displaystyle \int_{1}^{8} \frac{ e^{\sqrt[3]{x}} }{ \sqrt[3]{x^2} } \, dx$$

1. $$\int_{0}^{\pi} \cos(x+\pi) \, dx$$

2. Oil is leaking out of Enbridge Pipeline 5 under the Mackinac Bridge into Lake Michigan and Lake Huron at a rate of $$r(t) = 50 e^{-0.02t}$$ thousand liters per minute. How many liters leak out during the first hour?

## Thinky-thinky questions

1. Suppose $$\int_0^2 g(t) \, dt = 5$$.

1. Find $$\int_0^4 g(t/2) \, dt$$

2. Find $$\int_0^2 g(2-x) \, dx$$

2. Calculate exactly:

1. $$\int_{-\pi}^{\pi} \cos^2(\theta) \sin(\theta) \, d\theta$$.

2. Find the exact area under the curve $$y = \cos^2(t)\sin(t)$$ on $$0 \leq t \leq \pi$$

## Thinky-thinky questions

Consider $$\int \cos(\theta) \sin(\theta) \, d\theta$$.

1. Evaluate the integral.

2. You probably used either $$u = \cos(\theta)$$ or $$u = \sin(\theta)$$. Use the other substitution to solve the integral.

3. Use a trig identity such as $$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$ or $$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$$ to solve the integral.

4. You should have found three expressions for $$\int \cos(\theta) \sin(\theta) \, d\theta$$. Are they really different? Are they all correct? Explain.