Bill Foster

Solve for $x$:  $\log_{a}(x+b)- \log_{a}(x+c)=d$

Find a regression equation.

Find a regression equation.

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$. Solution involves the inverse trigonometric function $\arcsin$.

Find $\displaystyle \int\frac{ax+b}{(1-x^2)^{1/2}} \;dx$. Solution involves inverse trigonometric functions.

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$

Find $\displaystyle \int (ax+b)\cos(cx+d)\; dx $ and hence find $\displaystyle \int (ax+b)^2\sin(cx+d)\; dx $ 

Also two other questions on integrating by parts.

Find the polynomial $g(x)$ such that $\displaystyle \int \frac{ax+b}{(cx+d)^{n}} dx=\frac{g(x)}{(cx+d)^{n-1}}+C$.

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$.

Find $\displaystyle \int ax ^ m+ bx^{c/n}\;dx$.

Paired t-test to see if there is a difference between times take in a task.

Two sample t-test to see if there is a difference between scores on questions between two groups when the questions are asked in a different order.

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$

Find  $\displaystyle \int\cosh(ax+b)\;dx,\;\;\int x\sinh(cx+d)\;dx$

Differentiate the following functions: $\displaystyle x ^ n \sinh(ax + b),\;\tanh(cx+d),\;\ln(\cosh(px+q))$

Solve for $x$: $a\cosh(x)+b\sinh(x)=c$. There are two solutions for this example.

Algebraic manipulation/simplification.

Simplify $\displaystyle \frac{ax^4+bx^2+c}{a_1x^4+b_1x^2+c_1}$ by cancelling a a common degree 2 factor.

Solve for $x$: $\displaystyle ax ^ 2 + bx + c=0$.

Express a sum of linear terms in $x$ and $y$ as a single linear term in $x$ and $y$.

Expand $(az^2+bz+c)(dz^2+pz+q)$.

Expand $(az^2+bz+c)(pz+q)$.

Expand $(pw+q)(aw^2+bw+c)$.

Expand $(ax+b)(cx+d)$.

Expand $ax(cx+d)$ and expand $(rx+s)(px)$

Expand $(ay+b)(cy+d)(py+q)$.

Includes a video expanding three brackets, however uses the variable $x$ rather than $y$.

Simplify $(ax+b)(cx+d)-(ax+d)(cx+b)$. Answer is a multiple of $x$.

Simplify $(ax+by)(cx+dy)-(ax+dy)(cx+by)$. Answer is a multiple of $xy$.