The trick here is to use an appropriate substitution. Let u=a^3.
Then du/da=3a^2, and du=3a^2da.
We can now make two key substitutions: In (3a^2)da/(1+a^6), replace 3a^2 by du and a^6 by u^2.
Then we have the integral of du/(1+u^2).
Integrating, we get arctan u + c. Substituting a^3 for u, the final result (the integral in question) is arctan a^3 + c.
Check this by differentiation. if you find the derivative with respect to a of arctan a^3 + c, you MUST obtain the result 3a^2/(1+a^6).
2^x * 2 = 15
2^x= 15/2
2^x = 7,5
x = log 2 (7,5)
I think the answer is C because you would A=Pi x r x 2
Since he must not exceed 8 hours driving in a day:
Let the distance of the picnic be = x km.
Therefore time for forward journey = x / 60
Return journey = x / 50
The total trip should not exceed 8 hours.
Therefore: x / 60 + x / 50 <= 8. LCM = 300
Taking LCM and multiplying on both sides:
5x + 6x <= 8(300)
11x <= 2400
x <= 2400/11
x <= 218.18
The picnic spot must be less than or equal to 218.18 km.
