Your work is fine. It looks like your professor didn't substitute for dt.mk200789 said:Homework Statement
integrate (x^3)sqrt(x^2 + 8) dx
Homework Equations
The Attempt at a Solution
let x = 2sqrt(2)tan(t) ==> dx= 2sqrt(2)sec^2(t) dt
=int (x^3)sqrt(x^2 + 8) dx
=int (16sqrt(2)tan^3(t)) sqrt(8tan^2(t) + 8) (2sqrt(2)sec^2(t)) dt
=int (32 sqrt(2)) (tan^3(t)) sec^3(t) dt
my problem is till this point i compared it with the the prof. working
and he had this instead : int (16sqrt(2)tan^3(t))(2sqrt(2)sec(t)) dt
I didn't really take a look at your solution
But in order to solve the integral ,, just substitute >>> u = 8 + x^2
it's straight forward ;)
Cyosis said:It may be straightforward but this substitution is pretty useless.
Integration by parts is the way to go here and you only have to do it once.
The solution comes out almost straight away with the substitution actually.
Gib Z said:The solution comes out almost straight away with the substitution actually.
Char.Limit said:How exactly? du does not equal x^3, it equals 2x. Rather useless.
mk200789 said:SO...my working so far is correct? or the prof's correct?:|
=int (16sqrt(2)tan^3(t)) sqrt(8tan^2(t) + 8) (2sqrt(2)sec^2(t)) dt
=int (32 sqrt(2)) (tan^3(t)) sec^3(t) dt
No integration by parts is needed with the substitution. You just multiply it out to get two terms of the form u^n.Cyosis said:While useless may have been a bit strong I still feel you don't gain much by doing that substitution. You will have to do integration by parts after wards, which you may as well do right away.
The purpose of integrating (x^3)sqrt(x^2 + 8) dx is to find the area under the curve of the given function. Integration is a mathematical process used in calculus to find the total accumulation of a quantity over a given interval.
To solve this integral, we can use the substitution method. Let u = x^2 + 8, then du = 2x dx. Substituting this into the integral, we get 1/2 * ∫ u^(3/2) du, which can be easily evaluated as 1/2 * (2/5)u^(5/2) + C. Substituting back for u, we get the final solution of 1/5 * (x^2 + 8)^(5/2) + C.
The domain of this function is all real numbers, as there are no restrictions on the values that x can take.
Yes, there is a shortcut method known as the power rule for integration. In this case, we can use the power rule by adding 1 to the power of x and dividing the coefficient by the new power. The integral then becomes ((x^3)/4) * (x^2 + 8)^(3/2) + C.
You can check your solution by taking the derivative of your answer. If the derivative is equal to the original function, then your solution is correct. In this case, the derivative of 1/5 * (x^2 + 8)^(5/2) + C is indeed equal to (x^3)sqrt(x^2 + 8).