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## Relations and Functions **RR** given by $f(x)$ = ### **Positive or Negative** $x$ if $x > 0$ 0 if $x = 0$ $-x$ if $x < 0$ **RR** given by $f(x) = -1$ if $x0$ And let $f$ be a function. **State whether the following is injective, onto, or bijective.** 1. Consider the function $f(x) = x$....

## Relations and Functions **RR** given by $f(x)$ = ### **Positive or Negative** $x$ if $x > 0$ 0 if $x = 0$ $-x$ if $x < 0$ **RR** given by $f(x) = -1$ if $x0$ And let $f$ be a function. **State whether the following is injective, onto, or bijective.** 1. Consider the function $f(x) = x$. 2. Consider the function $f(x) = x$. 3. Consider the function $f(x) = 3x$. **Integrations** 1. $\int constant \cdot dx = x + c$ 2. $\int(f(x) + g(x)) dx = \int f(x)dx + \int g(x) dx$ 3. $\int k f(x) dx = k \int f(x) dx$ **Substitution** 1. $\int (ax + b) ^n dx = \frac {1}{a} \frac {(ax + b)^{n+1}}{n+1} + c$ 2. $\int Sec^2 (ax + b)dx = \frac {1}{a} Tan(ax + b) + c$ 3. $\int sin (ax + b) dx = -\frac {1}{a} Cos(ax + b) + c$ 4. $\int cos (ax + b) dx = \frac {1}{a} sin(ax + b) + c$ 5. $\int cosec^2 (ax + b)dx = -\frac {1}{a} Cot(ax + b) + c$ 6. $\int sec(ax + b)tan(ax + b) dx = \frac {1}{a} sec(ax + b) + c$ 7. $\int cosec(ax + b)cot(ax + b)dx = -\frac {1}{a} cosec(ax + b) + c$ ### Model problems 1. $\int cos(x + a) \cdot cos (x + b) dx$ *Multiply & subtract* $\int cos(x + a)\cdot cos (x + b)dx$ = $\frac{1}{2} sin(a - b) + \frac{1}{2}$ $\int cos(2x + a + b)dx$ 2. When $x = \tan(x+a)$, then