Question

Evaluate the indefinite integral.

Answer 1

x+c

Answer 2

\[\int\limits x(2x+5)^{8}dx\]

Answer 3

what exponent is that?

Answer 4

8

Answer 5

Seems like a parts question?

Answer 6

well let me ask you a question, how would you approach this?

Answer 7

can i put u=2x+5

Answer 8

Naw

Answer 9

what can i do then?

Answer 10

integration by parts:

\[\int\limits_{}^{} u dv = uv - \int\limits_{}^{} v du\]

in this case

u = x and dv = (2x + 5)^8

therefore du = dx and v = (2x+5)^9/18

now that you have all the components, u, v and du. you can solve using the equation.

so your integral is equivalent to:\[\frac{ x(2x+5)^9 }{ 18 } - \frac{ 1 }{ 18 } \int\limits_{}^{}(2x+5)^9 dx\]

Answer 11

you can solve the second one

Answer 12

why is it 18?

Answer 13

If for some reason you wanted to avoid using Integration by Parts, here is an approach:\[\Large\bf\sf \int\limits x(2x-5)^8\;dx\]
\[\Large\bf\sf u=2x-5 \qquad\to\qquad x=\frac{1}{2}u+\frac{5}{2}\]\[\Large\bf\sf \frac{1}{2}du=dx\]

Giving us:\[\Large\bf\sf \int\limits\limits \left(\frac{1}{2}u+\frac{5}{2}u\right)u^8\cdot\frac{1}{2}du\quad=\quad \frac{1}{2}\int\limits \frac{1}{2}u^9+\frac{5}{2}u^8\;du\]Which shouldn't be too hard to evaluate.

Answer 14

This method isn't necessarily easier ^
Just another option.

Answer 15

thank you @zepdrix