# FTC and Antiderivatives

Math 111

### That last term from the applet...

Why did it change?

### +C

Why the +C from the general antiderivative?

### We know FTC

Part 1: If $$v(t)$$ is a velocity function, we can write $$p(t)$$ as: $p(t)=\int_0^t v(x)dx$ Why the variables?

Part 2: If $$F'(x)=f(x)$$, $\int_a^b f(x)dx = F(b)-F(a)$

### Notation

We use $$\int f(x) dx$$ to denote the general antiderivative to $$f(x)$$

### Scenarios

• Coal Scrubbers: Pollution rate * $$\Delta$$ Time
• Mars Rover: Dust rate * $$\Delta$$ Distance
• Hydrostatic: Pressure * Width * $$\Delta$$ Height
• Chicken: Distance rate * $$\Delta$$ Time

### Scenarios Units

• Coal Scrubbers: Tons/Month * $$\Delta$$ Months
• Mars Rover: Mg/km * $$\Delta$$ km
• Hydrostatic: Lbs/ft 2 * ft * $$\Delta$$ ft
• Chicken: miles/hour * $$\Delta$$ Hours

### Using the FTC

Evaluate:

$\int_{-1}^2 \sin(x) dx$

$\int_{-1}^2 \cos(x) dx$

$\int_{-1}^t \sin(x) dx$

$\int_3^4 \dfrac{1}{x} dx$

### General propoerties

$\int_a^b f(x) dx = -\int_b^a f(x) dx$

$\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$

What if $$a < c < b$$?

### Challenge

Find a general antiderivative to

$f(x)=e^{5x^2}$

### Notation

$\int \dfrac{1}{x} dx$ vs $\int \dfrac{1}{t} dt$ vs $\int \dfrac{1}{(cabin)} d(cabin)$

### Integral Rules

Note that the FTC means these are useful for both indefinite definite integrals.

• Monomials
• Polynomials
• Trig function
• log functions
• exponential functions

Note: products and compositions are going to be harder

### General strategies

• If you have a polynomial divided by $$x^a$$
• If you know a particular value, e.g. F(1)=4
• Derivative of integral vs integral of derivative

### Back to the applet

• How do we know the constant term in $$q(t)$$?
• How does it change when we change $$a$$? How can we figure out the value of the constant?