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?

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

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.

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

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 \[\begin{array}{rcl} \int_{x=a}^{x=b} F'(g(x)) g'(x) \, dx & = & \int_{u=g(a)}^{u=g(b)} F'(u) \, du \\ & = & F(g(b)) - F(g(a)). \end{array}\]

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. \(\displaystyle \int \frac{1}{9+x^2} \, dx\)

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. \(\displaystyle \int \frac{x + 2 \sqrt{x-1}}{ 2x \sqrt{x-1}} \, dx\)

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?

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 \(0 \leq t \leq \pi\)

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.