Question

Calcule as integrais definidas
:
a) integral 3 (3x²/5 + 7x)dx
-1

b) integral 4 (2x³ - 5x² + 1)dx
0

Answer 1

Olá


Integração de polinômios:


$\displaystyle\mathsf{\int x^pdx~=~ \frac{x^{p+1}}{p+1}+c }$



A)


$\displaystyle\mathsf{ \int\limits^3_{-1} { \left(\frac{3x^2}{5}+7x \right) } \, dx }\\\\\\\\\mathsf{= \left(\frac{3}{5}\cdot \frac{x^{2+1}}{2+1}~+~7\cdot \frac{x^{1+1}}{1+1}\right)\bigg|^3_{-1} }\\\\\\\\\mathsf{= \left(\frac{x^3}{5}~+~ \frac{7x^2}{2}\right)\bigg|^3_{-1} }\\\\\\\mathsf{= \left(\frac{(3)^3}{5}~+~ \frac{7(3)^2}{2}\right)~-~ \left(\frac{(-1)^3}{5}~+~ \frac{7(-1)^2}{2}\right)}\\\\\\\mathsf{ =\frac{369}{10}~-~ \frac{33}{10} }\\\\\\\boxed{\mathsf{= \frac{168}{5} }}$



B)

$\displaystyle\mathsf{ \int\limits^4_{0} { \left(2x^3-5x^2+1 \right) } \, dx }\\\\\\\\\mathsf{= \left(2\cdot \frac{x^{3+1}}{3+1}~-~5\cdot \frac{x^{2+1}}{2+1}~+~x\right)\bigg|^4_{0} }\\\\\\\\\mathsf{= \left(\frac{x^4}{2}~-~ \frac{5x^3}{3}~+~x\right)\bigg|^4_{0} }\\\\\\\mathsf{= \left(\frac{(4)^4}{2}~-~ \frac{5(4)^3}{3}~+~(4)\right)~-~\left(\frac{(0)^4}{2}~-~ \frac{5(0)^3}{3}~+~(0)\right)}\\\\\\\mathsf{ =\frac{76}{3}~-~0}\\\\\\\boxed{\mathsf{= \frac{76}{3} }}$