Using an upper-case "C" for any arbitrary constants, find the general indefinite integral
∫ (-2-t)(-9-t^2) dt
Now I multiplied both parentheses to get:
∫ (18 + 2t^2 + 9t + t^3) dt
now I integrated and got:
18t + 2t^3/3 + 9t^2/2 + t^4/4 + C
but I'm told this answer is incorrect.
What am I doing wrong?
Thank you
I have
(t^2 + 9)(t+2)
is
t^3 + 2 t^2 + 9 t + 18
t^4/4 + (2/3) t^3 + (9/2) t^2 + 18 t + C
I agree with you.
okay so there must be a mistake with the system, thanks for clearing that up Damon.
Well, it seems like you've multiplied the parentheses correctly and integrated the resulting terms. But remember, when you integrate a polynomial, you need to divide each term by the power of the variable plus one.
So, let's go through it step by step. The integral of the expression (18 + 2t² + 9t + t³) dt can be found by integrating each term separately:
∫ 18 dt = 18t
∫ 2t² dt = (2/3)t³
∫ 9t dt = (9/2)t²
∫ t³ dt = (1/4)t⁴
Now, we add them up:
18t + (2/3)t³ + (9/2)t² + (1/4)t⁴ + C
And that should be the correct answer. Make sure you divide each term by the appropriate power of the variable. Sometimes it's easy to miss those little details, especially if you're under a lot of pressure. Remember, even mathematicians need a good laugh from time to time!
You made a small mistake while integrating the term t^3. When integrating t^3, you should divide by 4, not multiply by 4. So the correct integration would be:
∫ (18 + 2t^2 + 9t + t^3) dt = 18t + 2t^3/3 + 9t^2/2 + t^4/4 + C
So, the error is in the term t^3, where you should have t^3/3 instead of t^3.
The correct answer should be:
18t + 2t^3/3 + 9t^2/2 + t^4/4 + C
To find the general indefinite integral ∫ (-2 - t)(-9 - t^2) dt, you correctly multiplied the parentheses and obtained ∫ (18 + 2t^2 + 9t + t^3) dt. However, when you integrated, there seems to be an error. The correct integration is as follows:
∫ (18 + 2t^2 + 9t + t^3) dt
To integrate each term, you need to use the power rule for integration. The power rule states that the integral of x^n with respect to x, where n is a constant, is (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule to each term, we have:
18t (using the power rule with n = 0)
2t^3/3 (using the power rule with n = 2)
9t^2/2 (using the power rule with n = 1)
t^4/4 (using the power rule with n = 3)
So the correct answer should be:
18t + 2t^3/3 + 9t^2/2 + t^4/4 + C
Make sure to double-check your integration steps to ensure that you correctly applied the power rule.