Calculus I H | 8.1 and 8.3 Review

 $\int e^{-x^4}\left(-4x^3\right)dx$  Find the antiderivative. 

•  $e^{-x^4}+C$  

•  $2e^{-x}+C$  

•  $e^{2x}+C$  

•  $\ln-x^4+C$  

 $\int\frac{e^{\frac{1}{x^2}}}{x^3}dx$  Find the antideriavtive.

•  $-\frac{1}{2}e^{\frac{1}{x^2}}+C$  

•  $-\frac{1}{4}e^x+C$  

•  $\frac{1}{2}e^{-\frac{1}{x^2}}-C$  

•  $-2e^{\sqrt{x}}-C$  

 $\int\ \frac{3}{x}dx$  

 $\int\ \frac{1}{x+1}dx$  Find antiderivative

 $\int\ \frac{x}{x^2+1}dx\$  Find the antiderivative

•  $\frac{1}{2}\ln\ \left|x^2+1\right|+C$  

•  $2\ln\ \left|x+1\right|+C$  

•  $\ln\ \left|x^2\right|+C$  

•  $-\frac{1}{4}\ln\ \left|x+2\right|+C$  

 $\int\ \frac{x^3-3x^2+5}{x-3}dx$  

Divide using long division, then integrate

•  $\frac{x^3}{3}\ +\ 5\ln\left|x-3\right|+C$  

•  $\frac{x^2}{2}\ +\ \ln\left|x-2\right|+C$  

•  $\frac{x}{4}\ +\ 7\ln\left|x-5\right|+C$  

•  $\frac{x^8}{8}\ +\ 7\ln\left|x-2\right|+C$  

 $\int3^xdx$  Find the indefinite integral

•  $\frac{1}{\ln3}\cdot3^x+C$  

•  $3^x+C$  

•  $e^3\cdot3^{ }+C$  

•  $\frac{1}{3x}\cdot3^x+C$  

 $\int x\left(5^{-x^2}\right)dx$  Find antidereivative

•  $-\ \frac{1}{2\ln5}\cdot5^{-x^2}+C$  

•  $\ \frac{1}{2\ln}\cdot2^{x^5}+C$  

•  $-\ \frac{1}{2}\cdot5^{x^2}+C$  

•  $\ \frac{1}{e^{2x}}\cdot2^x+C$  

What are some accurate rules for relating to Ln and e anti-derivatives?

• $y\ =\ \ln\left(u\right)+C\$ ↔️ $\frac{dy}{dx}\ =\ \frac{1}{u}\cdot du$

• $e^udu$ ↔️ $e^u+C$

• $\int e^uax$ ↔️ $a\int e^ux$ (where 'a' is a constant)

• $\int e^ux$ ↔️ $x\int e^u$

• $y\ =\ a^x$ ↔️ $\frac{dy}{dx}=\ln a^x$

 $\frac{x^2+x+1}{x+3}$  Long division practice: Use long division with polynomials to find answer.

•  $x-2+\frac{7}{x+3}$  

•  $x+3$  

•  $x+3+\frac{8}{x+3}$  

•  $x^2+x-1+\frac{9}{x+3}$