Answer:
A)2
Step-by-step explanation:
we would like to integrate the following definite Integral:
[tex] \displaystyle \int_{0} ^{1} 5x \sqrt{x} dx[/tex]
use constant integration rule which yields:
[tex] \displaystyle 5\int_{0} ^{1} x \sqrt{x} dx[/tex]
notice that we can rewrite √x using Law of exponent therefore we obtain:
[tex] \displaystyle 5\int_{0} ^{1} x \cdot {x}^{1/2} dx[/tex]
once again use law of exponent which yields:
[tex] \displaystyle 5\int_{0} ^{1} {x}^{ \frac{3}{2} } dx[/tex]
use exponent integration rule which yields;
[tex] \displaystyle 5 \left( \frac{{x}^{ \frac{3}{2} + 1 } }{ \frac{3}{2} + 1} \right) \bigg| _{0} ^{1} [/tex]
simplify which yields:
[tex] \displaystyle 2 {x}^{2} \sqrt{x} \bigg| _{0} ^{1} [/tex]
recall fundamental theorem:
[tex] \displaystyle 2 ( {1}^{2}) (\sqrt{1} ) - 2( {0}^{2} )( \sqrt{0)} [/tex]
simplify:
[tex] \displaystyle 2 [/tex]
hence
our answer is A
