Question

\(\int_0^1 {\left( {1 - {x^9}} \right)dx} {\rm{ }}\)

Short answer

Integrate the function,

\(\begin{aligned}{c}\int_0^1 {\left( {1 - {x^9}} \right)dx} {\rm{ }} &= \left. {\left( {x - \frac{{{x^{10}}}}{{10}}} \right)} \right|_0^1\\ &= \left( {\frac{9}{{10}}} \right) - 0\\ &= \frac{9}{{10}}\end{aligned}\)

Step-by-step solution

Use the difference rule for integration

\(\int_0^1 {\left( {1 - {x^9}} \right)dx} = \int_0^1 {dx} - \int_0^1 {{x^9}dx} \)

Use the integral formula \(\int_1^2 {{x^n}dx}  = \frac{{{x^{n + 1}}}}{{n + 1}} + C\)

\(\left. { = \left( {x - \frac{{{x^{10}}}}{{10}}} \right)} \right|_0^1\)

Use the integral formula \(\int_a^b {f\left( x \right)dx}  = F\left( b \right) - F\left( a \right)\)

\(\begin{aligned} &= \left( {1 - \frac{{{1^{10}}}}{{10}}} \right) - 0\\ &= \left( {\frac{9}{{10}}} \right) - 0\\ &= \frac{9}{{10}}\end{aligned}\)

Therefore, the value is \(\frac{9}{{10}}\).