a) Calcular la derivada de 
b) Calcular
c) Calcular
Solución:
a) Recordar la tabla de derivadas:
b) Recordamos que:
Luego:
c) Calculamos la integral:
![\displaystyle\int_0^2\left(3x^2-\dfrac1{\sqrt{4x+1}}\right)~dx=\int_0^2\left(3x^2-(4x+1)^{-1/2}\right)~dx=\\\\=\int_0^2\left(3x^2-\dfrac14\cdot(4x+1)^{-1/2}\cdot4\right)~dx=\left[\dfrac{3x^3}3-\dfrac14\cdot\dfrac{(4x+1)^{1/2}}{1/2}\right]_0^2=\\\\=\left[x^3-\dfrac{\sqrt{4x+1}}2\right]_0^2=\left(2^3-\dfrac{\sqrt{4\cdot2+1}}2\right)-\left(0-\dfrac12\right)=\\\\=8-\dfrac32+\dfrac12=\boxed{7}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cint_0%5E2%5Cleft%283x%5E2-%5Cdfrac1%7B%5Csqrt%7B4x%2B1%7D%7D%5Cright%29%7Edx%3D%5Cint_0%5E2%5Cleft%283x%5E2-%284x%2B1%29%5E%7B-1%2F2%7D%5Cright%29%7Edx%3D%5C%5C%5C%5C%3D%5Cint_0%5E2%5Cleft%283x%5E2-%5Cdfrac14%5Ccdot%284x%2B1%29%5E%7B-1%2F2%7D%5Ccdot4%5Cright%29%7Edx%3D%5Cleft%5B%5Cdfrac%7B3x%5E3%7D3-%5Cdfrac14%5Ccdot%5Cdfrac%7B%284x%2B1%29%5E%7B1%2F2%7D%7D%7B1%2F2%7D%5Cright%5D_0%5E2%3D%5C%5C%5C%5C%3D%5Cleft%5Bx%5E3-%5Cdfrac%7B%5Csqrt%7B4x%2B1%7D%7D2%5Cright%5D_0%5E2%3D%5Cleft%282%5E3-%5Cdfrac%7B%5Csqrt%7B4%5Ccdot2%2B1%7D%7D2%5Cright%29-%5Cleft%280-%5Cdfrac12%5Cright%29%3D%5C%5C%5C%5C%3D8-%5Cdfrac32%2B%5Cdfrac12%3D%5Cboxed%7B7%7D&bg=ffffff&fg=141412&s=0&c=20201002)
♦
a) Calcular la derivada de
b) Calcular
c) Calcular
Solución:
a) Recordar la tabla de derivadas:
b) Recordamos que:
Luego:
c) Calculamos la integral:
♦