Find what value of c does ∫[∞,9] c/(x^4) dx=1?
you mean for what value of c does
-c/2187 = 1
??
To find the value of c that satisfies the equation ∫[∞,9] c/(x^4) dx = 1, we need to evaluate the integral and solve for c.
Let's integrate the function ∫[∞,9] c/(x^4) dx.
First, we need to find the antiderivative (or indefinite integral) of c/(x^4). The antiderivative of c/(x^4) is -c/(3x^3).
So, the definite integral can be expressed as ∫[∞,9] c/(x^4) dx = -c/(3x^3) evaluated from ∞ to 9.
Now, let's evaluate the integral at the upper and lower limits:
∫[∞,9] c/(x^4) dx = (-c/(3*9^3)) - (-c/(3*∞^3))
The value of (∞^3) is infinity (∞), so the second term approaches zero.
The integral can be simplified to:
1 = (-c/(243)) - 0
Next, we can solve for c:
1 = -c/243
To isolate c, we can multiply both sides of the equation by -243:
-243 = c
Therefore, the value of c that satisfies the equation is c = -243.
To find the value of c that satisfies the equation ∫[∞,9] c/(x^4) dx = 1, we need to evaluate the integral on the left-hand side of the equation first and then solve for c.
Let's start by evaluating the integral. The integral of c/(x^4) dx can be found by using the power rule for integrals:
∫[∞,9] c/(x^4) dx = -c/(3x^3) + C,
where C is the constant of integration.
To evaluate this integral, we need to plug in the upper and lower limits of integration, which are ∞ and 9, respectively:
∫[∞,9] c/(x^4) dx = [-c/(3 * 9^3)] - [-c/(3 * ∞^3)] = -c/(3 * 729) - (-c/(3 * ∞^3)).
Now, we can set this result equal to 1 and solve for c:
1 = -c/(3 * 729) - (-c/(3 * ∞^3)).
Since division by infinity is undefined, we can disregard the term -c/(3 * ∞^3). Thus, our equation simplifies to:
1 = -c/(3 * 729).
To isolate c, we can multiply both sides of the equation by 3 * 729:
3 * 729 * 1 = -c.
Simplifying the left-hand side gives:
2187 = -c.
Finally, we solve for c by multiplying both sides of the equation by -1:
c = -2187.
Therefore, the value of c that satisfies the equation ∫[∞,9] c/(x^4) dx = 1 is c = -2187.