in the integral (tan^15(x)sec^8(x)dx
a substitution of u=sec^2(x) and the result is written as the integral f(u)du
find f(u)
I've gotten to the integral (tan^14(x)sec^6(x))/2 du but not sure where to go after
AAAaannndd the bot gets it wrong yet again!
∫ tan^15(x) sec^8(x) dx = ∫ tan^15(x) sec^6(x) sec^2(x) dx
Let u = tanx, du = sec^2x dx. Then you have
∫ u^15 (u^2+1)^3 du = 1/22 u^22 + 3/20 u^20 + 1/6 u^18 + 1/16 u^16
now just reverse the substitution and you're done
I dont think thats the answer! would appreciate another look at it
I tried entering the answer into my question but it still said it was wrong, also shouldnt u=sec^2(x) as per the question? I appreciate any help, been stuck on this for hours
I got the answer! for future readers, its ((u-1)^7u^3)/2
To find the expression f(u), we can start from where you left off:
The integral you have is ∫(tan^14(x)sec^6(x))/2 dx, after performing the substitution u=sec^2(x).
Now, let's express tan^14(x) in terms of sec(x) using the identity: tan^2(x) = sec^2(x) - 1.
So, tan^14(x) = (sec^2(x) - 1)^7.
Substituting this back into the integral, we get:
∫[(sec^2(x) - 1)^7 * sec^6(x)]/2 dx.
Now, using the identity sec^2(x) = 1 + tan^2(x), we can further simplify the integral:
∫[(1 + tan^2(x) - 1)^7 * sec^6(x)]/2 dx.
This simplifies to:
∫[tan^7(x) * sec^6(x)]/2 dx.
Now we can rewrite tan^7(x) in terms of sec(x) using the identity: tan^2(x) = sec^2(x) - 1.
tan^7(x) = (sec^2(x) - 1)^3 * tan(x).
Substituting this back into the integral, we get:
∫[(sec^2(x) - 1)^3 * tan(x) * sec^6(x)]/2 dx.
Now, let's rewrite sec^6(x) as (sec^2(x))^3:
∫[(sec^2(x) - 1)^3 * tan(x) * (sec^2(x))^3]/2 dx.
Finally, simplifying the expression further, we have:
∫[u^3 * tan(x)]/2 dx.
Therefore, the expression f(u) = u^3 * tan(x).
that.
Answer:
f(u) = (u^(7/2) tan^14(sec^-1(√u)))/2