∫9x dx=(coefficient)x(power)+c\displaystyle\int 9\sqrt{x} \ dx= ext{(coefficient)}x^{ ext{(power)}} +c
where coefficient = [Fill in the blank]

and power = [Fill in the blank]

(convert any fractional answers to decimal)

Question

∫9x dx=(coefficient)x(power)+c\displaystyle\int 9\sqrt{x} \ dx=	ext{(coefficient)}x^{	ext{(power)}} +c
where coefficient = [Fill in the blank]

and power = [Fill in the blank]

(convert any fractional answers to decimal)

BlackTom AI · Accepted Answer

Let’s parse the two integrals given and connect them to the requested form of an antiderivative.

First integral: ∫ 9x dx.
- In general, ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ -1. Here n = 1, so ∫ x dx = x^2/2.
- Multiply by the constant 9: ∫ 9x dx = 9 · (x^2/2) + C = (9/2) x^2 + C.
- This corresponds to a coefficient of 9/2 and a power of 2. Since the instruction says to convert fractional answers to decimal, 9/2 = 4.5, so the first integral would be: coefficient = 4.5, power = 2.

Second integral: ∫ 9√x dx = ∫ 9 x^{1/2} dx.
- Use the power rule with n = 1/2: ∫ x^{1/2} dx = x^{3/2}/(3/2) = (2/3) x^{3/2}.
- Multiply by 9: ∫ 9 x^{1/2} dx = 9 · (2/3) x^{3/2} + C = 6 x^{3/2} + C.
- This gives a coefficient of 6 and a power of 3/2, i.e., power = 1.5 when converted to decimal.

Now, addressing the provided answer values: ["6", "1.5"].
- These numbers precisely match the coefficient and power for the second integral, ∫ 9√x dx, after converting the fractional exponent to decimal (3/2 = 1.5).
- They do not match the first integral, which has coefficient 4.5 and power 2 when expressed in the requested form (convert 9/2 to decimal 4.5 and keep power 2).

Common misconceptions to watch out for:
- Confusing the two integrals and applying the same coefficient/power to both; each integral has its own antiderivative, and constants differ.
- Forgetting to apply the constant multiple rule correctly: multiplying the integral by 9 distributes to the coefficient 9 before applying the integral rule.
- Misapplying the power rule by incrementing the exponent incorrectly (n → n+1) or forgetting the division by (n+1).

In summary, the second integral yields coefficient 6 and power 1.5, which aligns with the provided values. The first integral would yield coefficient 4.5 and power 2, if that were being filled in separately.

∫9x dx=(coefficient)x(power)+c\displaystyle\int 9\sqrt{x} \ dx=\text{(coefficient)}x^{\text{(power)}} +c where coefficient = [Fill in the blank] and power = [Fill in the blank] (convert any fractional answers to decimal)

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