Question

resolver la integral ∫4(6+2x²)⁴dx​

Answer 1

Respuesta:

$\int \:4\left(6+2x^2\right)^4dx$

$\mathrm{Sacar\:la\:constante}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx$

$=4\cdot \int \left(6+2x^2\right)^4dx$

$Expandir}\:\left(6+2x^2\right)^4:\quad1296+1728x^2+864x^4+192x^6+16x^8$

$=4\cdot \int \:1296+1728x^2+864x^4+192x^6+16x^8dx$

$\mathrm{Aplicar\:la\:regla\:de\:la\:suma}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

$=4\left(\int \:1296dx+\int \:1728x^2dx+\int \:864x^4dx+\int \:192x^6dx+\int \:16x^8dx\right)$

$\int\:1296dx=1296x$

$\int\:1728x^2dx=576x^3$

$\int\:864x^4dx=\frac{864x^5}{5}$

$\int\:192x^6dx=\frac{192x^7}{7}$

$\int\:16x^8dx=\frac{16x^9}{9}$

$=4\left(1296x+576x^3+\frac{864x^5}{5}+\frac{192x^7}{7}+\frac{16x^9}{9}\right)$

RESULTADO:

$=4\left(1296x+576x^3+\frac{864x^5}{5}+\frac{192x^7}{7}+\frac{16x^9}{9}\right)+C$

Espero haberte ayudado :D