Question

integral problem

Answer 1

\[\int\limits_{0}^{\sqrt{x}} \ln (t+3^t)\]
find g'(x)

Answer 2

What variable is this being integrated with respect to

Answer 3

its probably t.

Answer 4

Want to make sure anyway

Answer 5

sorry dt

Answer 6

So you probably need to do integration by parts and use the fundamental theorem of calculus

Answer 7

can i use the u substitution rule?

Answer 8

i don't think so.

if you have u = t+3^t,

then you have du/dt = 1+3^t(log(3).

you keep getting 3^t back.

Answer 9

Yes, substitute u = ln(t + 3^t)

Answer 10

then what would my du be??
and the dx?

Answer 11

\[\Large g(x)=\int_0^{\sqrt x} \ln(t+3^t)\mathrm dt\]
use the fundamental theorem of calculus and the chain rule for differentiation

if \(\Large \displaystyle g(x)=\int_a^{u(x)} f(t) \mathrm dt\)

then \(\Large \displaystyle g'(x) = f(u(x))\cdot u'(x)\)

Answer 12

oh ok that makes sense

Answer 13

as to the problem, \[\Large g'(x)=\ln(\sqrt x +3^{\sqrt x}) \frac{d(\sqrt x)}{dx}\]