Find the particular antiderivative that satisfies the following conditions:
H′(x)=(9/x^2)−(5/x^6) ;H(1)=0.
Did you look at the solution I gave you a few post back
to the same type of problem??
To find the particular antiderivative that satisfies the given conditions, we need to integrate the function H'(x) and evaluate the constant of integration using the given initial condition.
Integrating H'(x) with respect to x, we get:
∫(9/x^2) - (5/x^6) dx
Let's evaluate each term separately:
∫(9/x^2) dx = 9∫(1/x^2) dx = 9 * -1/x = -9/x
∫(5/x^6) dx = 5∫(1/x^6) dx = 5 * -1/5x^5 = -1/x^5
Combining these results, the antiderivative of H'(x) is given by:
H(x) = -9/x - 1/x^5
To find the constant of integration, we can substitute the value H(1) = 0:
0 = -9/1 - 1/1^5
0 = -9 - 1
0 = -10
Therefore, the particular antiderivative that satisfies the given conditions is:
H(x) = -9/x - 1/x^5 + 10
To find the particular antiderivative of H'(x) that satisfies the given conditions, we can integrate H'(x) with respect to x and then use the initial condition to determine the constant of integration.
First, integrate H'(x) term by term. For each term, use the power rule of integration:
∫(9/x^2) dx = ∫9x^(-2) dx = -9x^(-1) = -9/x,
∫(5/x^6) dx = ∫5x^(-6) dx = 5/(-6+1) x^(-6+1) = -5/6x^(-5) = -5/(6x^5).
Next, combine the results:
H(x) = -9/x - 5/(6x^5).
Now, we can use the given initial condition H(1) = 0 to determine the constant of integration. Plug in x = 1 into the expression for H(x) and solve for the constant:
H(1) = -9/1 - 5/(6*1^5) = -9 - 5/6 = 0,
-9 - 5/6 = 0.
Multiply through by 6 to clear the fraction:
-54 - 5 = 0,
-59 = 0.
This equation is contradictory, indicating that there is no particular antiderivate of H'(x) that satisfies the given conditions.