Question

whats the result of this integral?

Answer 1

\[\int\limits_{}^{}t ^{2}(9t ^{2} + 1)\]

Answer 2

\[\int\limits\limits_{}^{}\sqrt{t ^{2}(9t ^{2} + 1)}\] I forgot that it had a root ... so do I have to do the same?

Answer 3

in this case, you want to take the t^2 out of the square root.

Answer 4

then you get \[\int\limits x*\sqrt(9x^2+1)\]

Answer 5

dx, of course

Answer 6

Yeah, now you can make make a regular sub, no need for trig actually.

Answer 7

Now, use u-substitution. What would be a good value of u?

Answer 8

Do we have a particular domain for \(t\) ? Recall that \(\sqrt{t^2}=|t|\), which is equivalent to \(t\) only if \(t\ge0\).

Answer 9

I suppose if we multiplied by |t|/t at the end, but that seems a bit inelegant. I don't see any great way to fix that inconsistency.

Answer 10

but how can I do the u substitution ? what formula do I have to apply?

Answer 11

@HarryPotter5777

Answer 12

You'll want to use u=9t^2+1

Answer 13

then you have sqrt(u)*t. And t can be expressed in terms of du. What's the derivative of du?

Answer 14

*the derivative of u, sorry.

Answer 15

is:18t

Answer 16

right. Then you express in in terms of u: sqrt(u)du/18. Now integrate, and you're done! Just remember +C.