Evaluate:
integral x^3/[sqrt*(x^2+100)]dx
can someone tell me what they get as a final answer..I've been going at this for hours and don't get the answer in the back of the textbook
Answer
1/3 (sqrt(x^2 + 100))^3 - 100(sqrt(x^2+100)) + C
this can be transformed into
1/3 (x^2 - 200)(sqrt(x^2 + 100) + C
my book had 1st answer (not transformed)
let me know if you need more help with this
forgot to add, there are other forms for this answer
post your answer if you are not sure if equil. to above posted answers
Excellent !
I got the answer your book had!
Thankyou so much !!
To evaluate the integral ∫ (x^3 / √(x^2 + 100)) dx, we can use the substitution method.
Let's set u = x^2 + 100. This means that du = 2x dx, or dx = du / (2x). Substituting these expressions into the integral, we have:
∫ (x^3 / √(x^2 + 100)) dx = ∫ (x^3 / √u) (du / (2x))
= (1/2) ∫ (x^2 / √u) du
Now, let's focus on the remaining integral. We can rewrite x^2 as x * x. Since x * x = u - 100, we can substitute this expression into the integral:
(1/2) ∫ ((u - 100) / √u) du
To simplify this integral further, let's split it into two separate integrals:
(1/2) ∫ (u/√u) du - (1/2) ∫ (100/√u) du
Integrating each term separately, we get:
(1/2) ∫ (√u) du - (1/2) ∫ (100/√u) du
Let's simplify these integrals:
(1/2) * (2/3) * u^(3/2) - (1/2) * (100) * (2) * (√u)
Simplifying further, we have:
(u^(3/2) / 3) - 100√u + C
Now, substitute back u = x^2 + 100 to get the final answer:
(x^2 + 100)^(3/2) / 3 - 100√(x^2 + 100) + C
Please note that the result may differ depending on the initial limits of integration. Make sure to apply them to the final answer accordingly.