(a) Integrate wrt x:(〖2x〗^4- x^2+2)/x^2 (b) The gradient of the tangent at a point on a curve is given by x2 + 2x – 2. Find the equation of the curve if it passes through (2,3).
(a) To integrate the expression (2x^4 - x^2 + 2)/x^2 with respect to x, you can follow these steps:
Step 1: Rewrite the expression as the sum of separate fractions:
(2x^4 - x^2 + 2)/x^2 = 2x^2/x^2 - x^2/x^2 + 2/x^2
Step 2: Simplify each fraction:
= 2 - 1 + 2/x^2
Step 3: Split the integral into separate terms:
∫(2 - 1 + 2/x^2) dx = ∫2 dx - ∫1 dx + ∫2/x^2 dx
Step 4: Integrate each term individually:
∫2 dx = 2x + C1
∫1 dx = x + C2
∫2/x^2 dx = -2/x + C3
Step 5: Combine the results:
The final integrated expression is:
∫(2x^4 - x^2 + 2)/x^2 dx = 2x - x + (-2/x) + C, where C = C1 + C2 + C3
(b) To find the equation of a curve when given the gradient of the tangent and a point it passes through, you need to integrate the gradient function.
Step 1: Write down the given gradient function:
f'(x) = x^2 + 2x - 2
Step 2: Integrate the gradient function:
∫(x^2 + 2x - 2) dx = (1/3)x^3 + x^2 - 2x + C
Step 3: Substitute the x and f(x) values from the given point (2,3) into the integrated equation:
3 = (1/3)(2)^3 + (2)^2 - 2(2) + C
Step 4: Solve for the constant C:
3 = (8/3) + 4 - 4 + C
3 = 8/3 + C
Step 5: Simplify and find the value of C:
9/3 - 8/3 = C
1/3 = C
Step 6: Substitute the constant C back into the integrated equation:
f(x) = (1/3)x^3 + x^2 - 2x + 1/3
Therefore, the equation of the curve is f(x) = (1/3)x^3 + x^2 - 2x + 1/3.