Question

integral (1-2xcube)xsquare​

Answer 1

Given,

$\displaystyle\longrightarrow I=\int\left(1-2x^3\right)x^2\ dx$

Distributing $x^2$ inside,

$\displaystyle\longrightarrow I=\int\left(1\cdot x^2-2x^3\cdot x^2\right)\ dx$

$\displaystyle\longrightarrow I=\int\left(x^2-2x^5\right)\ dx$

Giving integral sign to each term,

$\displaystyle\longrightarrow I=\int x^2\ dx-\int2x^5\right)\ dx$

Taking constant out of the integral,

$\displaystyle\longrightarrow I=\int x^2\ dx-2\int x^5\right)\ dx$

Since $\displaystyle\int x^n\ dx=\dfrac{x^{n+1}}{n+1},$

$\displaystyle\longrightarrow I=\dfrac{x^{2+1}}{2+1}-2\cdot\dfrac{x^{5+1}}{5+1}+C$

$\displaystyle\longrightarrow I=\dfrac{x^{3}}{3}-2\cdot\dfrac{x^{6}}{6}+C$

$\displaystyle\longrightarrow I=\dfrac{1}{3}\,x^3-\dfrac{1}{3}\,x^{6}+C$

$\displaystyle\longrightarrow\underline{\underline{I=\dfrac{x^3}{3}\left(1-x^3\right)+C}}$

where $C$ is the integral constant.