U-Substitution
\int 2x \sqrt (4-x^2) dx
-\frac{2}{3} (4-x^2)^(\frac{2}{3}) + C
Find the u and dv of the integral:
\int\frac{ln(x)}{x^2}dx
u = ln(x), dv = \frac{1}{x^2}dx
\int sec(x)dx
ln|sec(x) + tan(x)| + C
\int \frac{1}{\sqrt(100-x^2)}dx
\arcsin(\frac{x}{9})+C
Write the partial fraction decomposition.
\frac{1}{(x+8)^3(x^2+2)}
\frac{Ax+B}{x^2+2}+\frac{C}{x+8}+\frac{D}{(x+8)^2}+\frac{E}{(x+8)^3}
Evaluate without a calculator.
2^11
2048
\int \cos^3\theta \sin\theta \d \theta
-\frac{1}{4} \cos^4\theta + C
\int \arctan(x)dx
xarctan(x)-\ln|x^2+1|+C
\int sin^2(x)cos^5(x)dx
\frac{1}{3}sin^3x-\frac{2}{5}sin^5x+\frac{1}{7}sin^7x + C
\int\frac{1}{\sqrt(49+x^2)dx
\ln|\frac{\sqrt(49+x^2)}{7}+\frac{x}{7}|+C
Write the partial fraction decomposition and solve for A and B.
\frac{3x+11}{x^2-x-6}
\frac{A}{x-3}+\frac{B}{x+2}
A=4, B=-1
What vegetable do you not want to bring on a boat?
a leak
\int 2x \sqrt (4-x^2) dx
\frac{16}{3} - 2 \sqrt(3)
\int \xtan^2(x)dx
-\frac{1}{2}x^2 + xtan(x) + \ln|\cos(x)|+C
\int sec^3(x) dx
\frac{1}{2}(secxtanx+ln|secx+tanx|)+C
\int\frac{x^3}{\sqrt(x^2-9))dx
27(\frac{\sqrt(x^2-9)}{3})+9(\sqrt(x^2-9)/3)^3+C
Write the partial fraction decomposition.
\frac{(x+3)}{x^2(x^4-1)(x^2+1)}
\frac{A}{x}+\frac{B}{x^2}+\frac{Cx+D}{x^2+1}+\frac{Ex+F}{(x^2+1)^2}+\frac{G}{x+1}+\frac{H}{x-1}
Two coins add up to 30 cents. One isn't a nickel. What are they?
A quarter and a nickel (only one isn't a nickel)
\int _0^{\frac{\pi }{2}}\sin (x)\cos (x)\sqrt{1-\sin (x)}dx
\frac{4}{15}
\int \sin^2(x)dx
\frac{1}{2}(-\sin(x)\cos(x)+x)+C
\int tan^7xsec^7xdx
\frac{1}{13}\sec^13(x)-\frac{3}{11}\sec^11(x)+\frac{1}{3}\sec^9(x)-\frac{1}{7}sec^7(x)+C
\int\frac{1}{\sqrt(x^2+2x)}dx
ln|(x+1)+\sqrt((x+1)^2-1)|+C
\int \frac{2x^2-x+4}{(x^2+4)(x-1)}dx
\frac{1}{2}\ln|x^2+4|+\ln|x-1|+C
It's the first thing you need with cosine. It's with students named Chris and Luca. But Maddy and Luis don't use it. It's the last thing to add when writing an integral. What is it?
C
\int \frac{x^3}{\sqrt{x^2-9}}dx
\frac{1}{3}(x^2-9)^\frac{3}{2} + 9 \sqrt{x^2-9} + C
\int ln(x)^2 dx
\xln(x)^2 - 2xln(x) + 2x + C
\int \frac{\cos^3x}{\sqrt(\sinx)dx
2\sqrt(\sinx)-\frac{2}{5}sin^(\frac{5}{2})x+C
\int(x-2)^3\sqrt(5+4x-x^2)dx
\frac{1}{5}(9-(x-2)^2)^(\frac{5}{2})-3(9-(x-2)^2)^(\frac{3}{2})+C
\int \frac{x-11}{(x^2+4)(x-1)}dx
\ln|x^2+4|+\frac{3}{2}\arctan(\frac{x}{2})-2\ln|x-1|+C
The man who invented it doesn’t need it. The man who bought it doesn’t want it. The man who needs it doesn’t know. What is it?
a coffin