Answer:
a) This integral can be evaluated using the basic integration rules. [tex]\int 11x^{4}dx = \frac{11}{5} x^{5}+C[/tex]
b) This integral can be evaluated using the basic integration rules. [tex]\int 8x^{1}x^{4}dx=\frac{4}{3}x^{6}+C[/tex]
c) This integral can be evaluated using the basic integration rules. [tex]\int 3x^{31}x^{4}dx=\frac{x^{36}}{12}+C[/tex]
Step-by-step explanation:
a) [tex]\int 11x^{4}dx[/tex]
In order to solve this problem, we can directly make use of the power rule of integration, which looks like this:
[tex]\int kx^{n}=k\frac{x^{n+1}}{n+1}+C[/tex]
so in this case we would get:
[tex]\int 11x^{4}dx=11 \frac{x^{4+1}}{4+1}+C[/tex]
[tex]\int 11x^{4}dx=11 \frac{x^{5}}{5}+C[/tex]
b) [tex]\int 8x^{1}x^{4}dx[/tex]
In order to solve this problem we just need to use some algebra to simplify it. By using power rules, we get that:
[tex]\int 8x^{1}x^{4}dx=\int 8x^{1+4}dx=\int 8x^{5}dx[/tex]
So we can now use the power rule of integration:
[tex]\int 8x^{5}dx=\frac{8}{5+1}x^{5+1}+C[/tex]
[tex]\int 8x^{5}dx=\frac{8}{6}x^{6}+C[/tex]
[tex]\int 8x^{5}dx=\frac{4}{3}x^{6}+C[/tex]
c) The same applies to this problem:
[tex]\int 3x^{31}x^{4}dx=\int 3x^{31+4}dx=\int 3x^{35}dx[/tex]
and now we can use the power rule of integration:
[tex]\int 3x^{35}dx=\frac{3x^{35+1}}{35+1}+C[/tex]
[tex]\int 3x^{35}dx=\frac{3x^{36}}{36}+C[/tex]
[tex]\int 3x^{35}dx=\frac{x^{36}}{12}+C[/tex]
