Integration

$\int_{ }^{ }\sin\ x\ \ dx\ =\ \cos\ x\ +c$

$\int_{ }^{ }e^x\ \ dx\ =\ \frac{e^{x+1}}{x+1}\ +\ c$

$\int_{ }^{ }\ \frac{1}{x}\ \ dx\ =\ \ln\ \left|x\ \right|\ +c$

$\int_{ }^{ }\ x\ \ dx\ =\ 1\ +\ c$

 $3x^2\ +\ c$  

 $\frac{x^4}{4}\ +\ c$  

 $\frac{x^4}{4}\ -\ \pi x$  

 $\frac{x^4}{4}\ -\ \pi x\ +\ c$  

 $5x^{-6}\ +\ c$  

 $5x^{-5}\ +\ c$  

 $-\frac{5}{4x^4}\ +\ c$  

 $-\frac{1}{x^5}\ +\ c$  

 $\sin\ \left(5x\right)\ +\ c$  

 $-\frac{1}{5}\sin\ \left(5x\right)\ +\ c$  

 $\frac{1}{5}\ \sin\ \left(5x\right)\ +\ c$  

 $\frac{\cos\ \left(5x\right)}{5}\ +\ c$  

 $3\left(5-3x\right)^2\ +c$  

 $\frac{\left(5-3x\right)^4}{4}+\ c$  

 $\frac{\left(5-3x\right)^4}{12}+\ c$  

 $-\frac{\left(5-3x\right)^4}{12}+\ c$  

Substitution

By Part

Quotient Rule

Partial Fraction

 $\frac{x^2}{2}+x$  

 $e^{2x}$  

 $2e^{2x}$  

 $\frac{e^{2x}}{2}$  

 $\ln\ \left|x\right|\ +\ 8\ \ln\left|2x-1\right|+c$  

 $\ln\ \left|x\right|\ +\ 4\ \ln\ \left|x-1\right|+c$  

 $\ln\ \left|x\right|\ +\ 4\ \ln\left|2x-1\right|+c$  

 $\ln\ \left|x\right|\ +\ 2\ \ln\left|2x-1\right|+c$  

 $7$  

 $8$  

 $12$  

 $x^3$  

 $2$  

 $4$  

 $6$  

 $8$