Question

encontre as integrais das funções a seguir:
a) f(x)= 2x+2
b) f(x)= 30x²+10x
c) f(x)= 3x²+2x-2
d) f(x)= x⁵
e) f(x)= 12x³- 15x²-2x+4​

Answer 1

$\large\boxed{\begin{array}{l}\sf a)~\rm f(x)=2x+2\\\displaystyle\rm\int (2x+2)\,dx=2\int x\,dx+2\int dx=2\cdot\dfrac{x^{1+1}}{1+1}+2x+c\\\displaystyle\rm\int (2x+2)\,dx=\dfrac{2}{3}x^3+2x+c\end{array}}$

$\large\boxed{\begin{array}{l}\sf b)~\rm f(x)=30x^2+10x\\\displaystyle\rm\int (30x^2+10x)\,dx=30\int x^2\,dx+10\int x\,dx\\\displaystyle\rm\int (30x^2+10x)\,dx=30\cdot\dfrac{x^{2+1}}{2+1}+10\dfrac{x^{1+1}}{1+1}+c\\\displaystyle\rm\int(30x^2+10x)\,dx=\dfrac{30}{3}x^3+\dfrac{10}{2}x^2+c\\\displaystyle\rm\int (30x^2+10x)\,dx=10x^3+5x^2+c\end{array}}$

$\large\boxed{\begin{array}{l}\sf c)~\rm f(x)=3x^2+2x-2\\\displaystyle\rm\int (3x^2+2x-2)\,dx=3\int x^2\,dx+2\int x\,dx-2\int dx\\\displaystyle\rm\int(3x^2+2x-2)\,dx=3\cdot\dfrac{x^{2+1}}{2+1}+2\dfrac{x^{1+1}}{1+1}-2x+c\\\displaystyle\rm\int(3x^2+2x-2)\,dx=\dfrac{3}{3}x^3+\dfrac{2}{2}x^2-2x+c\\\displaystyle\rm\int (3x^2+2x-2)\,dx=x^3+x^2-2x+c\end{array}}$

$\large\boxed{\begin{array}{l}\sf d)~\rm f(x)=x^5\\\displaystyle\rm\int x^5\,dx=\dfrac{x^{5+1}}{5+1}+c\\\displaystyle\rm\int x^5\,dx=\dfrac{1}{6}x^6+c \end{array}}$

$\large\boxed{\begin{array}{l}\sf e)~\rm f(x)=12x^3-15x^2-2x+4\\\displaystyle\rm\int(12x^3-15x^2-2x+4)\,dx=\\\displaystyle\rm 12\int x^3\,dx-15\int x^2\,dx-2\int x\,dx+4\int dx\\\displaystyle\rm\int (12x^3-15x^2-2x+4)\,dx=\\\rm 12\cdot\dfrac{x^{3+1}}{3+1}-15\dfrac{x^{2+1}}{2+1}-2\dfrac{x^{1+1}}{1+1}+4x+c\\\displaystyle\rm\int(12x^3-15x^2-2x+4)\,dx=\\\rm\dfrac{12}{4}x^4-\dfrac{15}{3}x^3-\dfrac{2}{2}x^2+4x+c\\\\\displaystyle\rm\int(12x^3-15x^2-2x+4)\,dx=3x^4-5x^3-x^2+4x+c\end{array}}$