Question

Starting with F(x)=∫(-2 to x) 3t^2(cos(t^3)+2)dt, use the substitution u(t)=t^3 to rewrite the definite integral.

Answer 1

\(\large F(x) = \int \limits_{-2}^x 3t^2 (\cos(t^3) + 2)dt\)

Answer 2

substitute \(\large u = t^3\)

Answer 3

\(\large du = 3t^2 dt\)

Answer 4

next work the bounds,
do u knw how to figure out the new bounds for \(u\) ?

Answer 5

thank you for replying! I'm not completely sure, can you please help?

Answer 6

the original bounds are (-2, x) right ?

Answer 7

when u substitute u = t^3,
the bounds change to :

((-2)^3, x^3)

(-8, x^3)

Answer 8

so the integral becomes :-

\(\large F(x) = \int \limits_{-8}^{x^3} (\cos(u) + 2) du\)

Answer 9

see if that makes some sense.. .

Answer 10

aha! I didn't realize how simple that was. thank you!

Answer 11

:) u wlc !