Compute $\int_{-1}^{1} 5x^{6}(1 - |x|^{5}) \mathop{dx}$

Question

Compute $$\int_{-1}^{1} 5x^{6}(1 - |x|^{5}) \mathop{dx}$$

$$\int_{-1}^{1} 5x^{6}(1 - |x|^{5}) \mathop{dx}$$ $$=\int_{-1}^{1} 5x^{6}- 5x^{6}|x|^{5} \mathop{dx} $$ $$= \int_{-1}^{1} 5x^{6} - 5\int_{-1}^{1}x^{6}|x|^{5}\mathop{dx}$$

$$= \frac{10}{7} - 5\left(\int_{-1}^{0}x^{6}|x|^{5} \mathop{dx} + \int_{0}^{1} x^{6}|x|^{5} \mathop{dx}\right) $$

$$= \frac{10}{7} -5\left(\int_{-1}^{0} x^{6}(-x)^{5} \mathop{dx} + \int_{0}^{1} x^{6} \cdot (x)^{5} \mathop{dx}\right) $$

$$= \frac{10}{7} - 5\left(\frac{1}{12} + \frac{1}{12}\right) $$

$$= \frac{25}{42}.$$

Is it ok ?

Answer 1

Another way:

As $x^6(1-|x|^5)$ is an even function,

$$I=\int_{-1}^15x^6(1-|x|^5)\ dx=2\int_0^15x^6(1-|x|^5)\ dx$$

$$\dfrac I{2\cdot5}=\int_0^1(x^6-x^{11})\ dx=?$$