Evaluate the following indefinite integral. Use C as the arbitrary constant.
�ç(12 /�ãx+12�ãx)dx
Only the x's are under the root symbols.
I got 6x^1/2 +8x^3/2+C as my answer. Is that correct?
I don't know, I can't read the original integral. But you can take the deriviative of your answer to check.
d/dx = 3/sqrtx + 12/sqrtx
That c symbol is suppose to be an integral symbol.
The a symbols are suppose to be square root symbols.
I don't know why that showed up like they did.
To evaluate the given indefinite integral, let's break it down step by step.
Given: ∫ (12 / √x + 12√x) dx
We can rewrite the expression as:
∫ (12 / x^(1/2) + 12x^(1/2)) dx
Now, to solve the integral, we can treat each term separately.
First term: ∫ 12 / x^(1/2) dx
To integrate 12 / x^(1/2), we apply the power rule for integration:
∫ x^n dx = (x^(n+1)) / (n+1), where n is not equal to -1
In this case, n = -1/2, so we have:
∫ 12 / x^(1/2) dx = 12 * (x^(-1/2 + 1)) / (-1/2 + 1) + C
= -24 * x^(1/2) + C
Second term: ∫ 12x^(1/2) dx
To integrate 12x^(1/2), we can use the power rule for integration again:
∫ x^n dx = (x^(n+1)) / (n+1), where n is not equal to -1
In this case, n = 3/2, so we have:
∫ 12x^(1/2) dx = 12 * (x^(3/2 + 1)) / (3/2 + 1) + C
= 12 * (x^(5/2)) / (5/2) + C
= (24/5) * x^(5/2) + C
Putting it all together, we get:
∫ (12 / √x + 12√x) dx = -24 * x^(1/2) + (24/5) * x^(5/2) + C
So, your answer of 6x^(1/2) + 8x^(3/2) + C is incorrect. The correct answer is:
-24 * x^(1/2) + (24/5) * x^(5/2) + C.