Evaluate the integral \(\int_{0}^{1}(8t^{3}i - 15t^{2}j + 20t^{4}k) dt\).

Solution:

Given,\(\int_{0}^{1}(8t^{3}i - 15t^{2}j + 20t^{4}k) dt\)

We have to evaluate the integral.

\(\\\int_{0}^{1}(8t^{3}i - 15t^{2}j + 20t^{4}k) dt\)

=\(\int_{0}^{1}8t^{3}i\, dt-\int_{0}^{1}15t^{2}j\, dt+\int_{0}^{1}20t^{4}k\, dt\\=8i\int_{0}^{1}t^{3}\, dt-15j\int_{0}^{1}t^{2}\, dt+20k\int_{0}^{1}t^{4}\, dt\\=8i\left [ \frac{t^{4}}4{} \right ]_{0}^{1}-15j\left [ \frac{t^{3}}{3} \right ]_{0}^{1}+20k\left [ \frac{t^{5}}{5} \right ]_{0}^{1}\\=(8i[(\frac{1^{4}}{4})-0])-(15j[\frac{1^{3}}{3}]-0)+(20k[\frac{1^{5}}{5}]-0)\\=(\frac{8i}{4})-(\frac{15j}{3})+(\frac{20k}{5})\\=2i-5j+4k\)

Therefore, the solution of the integral is 2i - 5j + 4k.

Evaluate the integral \(\int_{0}^{1}(8t^{3}i - 15t^{2}j + 20t^{4}k) dt\).

Summary:

On evaluating the integral \(\int_{0}^{1}(8t^{3}i - 15t^{2}j + 20t^{4}k) dt\) the solution is 2i - 5j + 4k.