Explain how you would use the power integration formula to integrate the function f(x) = (1/x^2)-(1/x)+(1/4).

Integrate each of the three terms separately, using what you call the "power integration formula", and add up the results.

The formula you are probably refering to is:

Integral of (a*x^n) = a*n*x^(n+1)/(n+1)

where a is the constant coefficient and n is the constant exponent.

1/4 can be thought of as (1/4)*x^0, so its integral is (1/4)*x^1/1 = x/4

The integral of the 1/x term is a special case, since you cannot divide by zero. Its integral is the natural logarithm of x, ln x

Now integrate the 1/x^2 term and add the integral results of all three terms. You can add an arbitrary constant at the end if you wish.

The final answer is

-1/x + ln x +x/4 + C

How do you integrate a radical function such as f(x)= �ã(7&4x^5 )- 3�ã(6&x^5 )- 11∜x +3∛x ? Describe your strategy.

To integrate the function f(x) = (1/x^2) - (1/x) + (1/4) using the power integration formula, you would follow a step-by-step process. Here's how you can do it:

Step 1: Identify the terms in the given function.
f(x) = (1/x^2) - (1/x) + (1/4)
Here, we have three terms: (1/x^2), -(1/x), and (1/4).

Step 2: Apply the power integration formula to each term.
The power integration formula states that ∫x^n dx = (x^(n+1))/(n+1) + C,
where C is the constant of integration.

Applying the power integration formula to each term, we get:
∫(1/x^2) dx = (x^(-2 + 1))/(-2 + 1) + C1 = -x^(-1) + C1
∫-(1/x) dx = -(x^(-1 + 1))/(-1 + 1) + C2 = -ln|x| + C2
∫(1/4) dx = (x^0)/(0+1) + C3 = x + C3

Step 3: Combine the integrals of each term.
∫f(x) dx = -x^(-1) + C1 - ln|x| + C2 + x/4 + C3

Step 4: Simplify the combined expression by combining the constants of integration.
Let's assume C = C1 + C2 + C3. Then, we can write:
∫f(x) dx = -x^(-1) - ln|x| + x/4 + C

Thus, the integrated form of f(x) using the power integration formula is:
∫f(x) dx = -x^(-1) - ln|x| + x/4 + C.

Remember to include the constant of integration (C) because it accounts for any possible change in the function's value.