Evaluate:
`a. int_1^2x^3dx` , b. `int_u^vmvdv`
c. `int_3^4(1/x)dx` , d. `int_4^9sqrtxdx`
e. `int_0^(pi//4)cos2xdx`

Explanation

a. `int_1^2x^3dx=(x^4/4)_1^2=2^4/4-1^4/4=15/4`
b. `int_u^vMvdv=(Mv^2/2)_u^v=M/2(v^2-u^2)`
c. `int_3^(4)1/xdx=(log_ex)_3^4=log_e4-log_e3=log_e(4//3)`
d. `int_4^9sqrtxdx=((x^(3//2))/(3//2))_4^9=2/3(9^(3//2)-4^(3//2))=38//3`
e. `int_0^(pi//4)cos2xdx=[(sin 2x)/(2)]_0^(pi//4)=1/2[sin{2(pi//4)}-sin(2xx0)]=1/2`