I'm trying to find the antiderivative of (x+3)/2
How would you go about this?
The answer is (x+3)^2/4
lets rewrite (x+3)/2 as:
(1/2)(x+3)^1
for antiderivatives (opposite of derivative), you add 1 from the exponent:
(1/2)(x+3)^2
then, you divide the coefficient (or 1/2) by the exponent (2):
(2)(1/2)(x+3)^2
so you'll end up with
(1/4)(x+3)^2
opps! grammatical error
add 1 to* the exponent
Wow, I knew it was easier than I thought. I tried splitting them up into x/2 + 3/2 and all kinds of things. Thanks so much.
To find the antiderivative of the expression (x+3)/2, we can use the power rule for integration. The power rule states that when integrating a term of the form x^n with respect to x, the result is (x^(n+1))/(n+1), where n is any real number except -1.
In this case, we have (x+3)/2. Let's simplify this expression first by dividing each term separately:
(x/2) + (3/2)
Now, let's apply the power rule to each term:
∫(x/2) dx = (1/2)∫x dx
Using the power rule, we increase the exponent of x by 1 and divide the term by the new exponent:
(1/2)(x^(1+1))/(1+1) = (1/2)(x^2)/2 = (x^2)/4
∫(3/2) dx = (3/2)∫1 dx
Since ∫1 dx is just the integral of a constant, it evaluates to x:
(3/2)x
Finally, putting the two terms together, we get the antiderivative:
(x^2)/4 + (3/2)x
This matches the given answer, (x+3)^2/4.
Therefore, the antiderivative of (x+3)/2 is (x^2)/4 + (3/2)x.