Question

\(\int_0^T {\left( {{x^4} - 8x + 7} \right)dx} {\rm{ }}\)

Short answer

Integrate the function,

\(\begin{aligned}{c}\int_0^T {\left( {{x^4} - 8x + 7} \right)dx} {\rm{ }} &= \left. {\left( {\frac{{{x^5}}}{5} - \frac{{8{x^2}}}{2} + 7x} \right)} \right|_0^T\\ &= \left( {\frac{{{T^5}}}{5} - 4{T^2} + 7T} \right) - 0\\ &= \frac{{{T^5}}}{5} - 4{T^2} + 7T\end{aligned}\)

Step-by-step solution

Use the sum and difference rule for integration

\(\int_0^T {\left( {{x^4} - 8x + 7} \right)dx} = \int_0^T {{x^4}dx} - \int_0^T {8xdx} + \int_0^T {7dx} \)

Use the constant rule for integration

\( = \int_0^T {{x^4}dx} - 8\int_0^T {xdx} + 7\int_0^T {dx} \)

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

\( = \left. {\left( {\frac{{{x^5}}}{5} - \frac{{8{x^2}}}{2} + 7x} \right)} \right|_0^T\)

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

\(\begin{array}{c} = \left( {\frac{{{T^5}}}{5} - \frac{{8{T^2}}}{2} + 7T} \right) - 0\\ = \left( {\frac{{{T^5}}}{5} - 4{T^2} + 7T} \right) - 0\\ = \frac{{{T^5}}}{5} - 4{T^2} + 7T\end{array}\)

Therefore, the value is \(\frac{{{T^5}}}{5} - 4{T^2} + 7T\).