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If y = x5 + 3x2, find the derivative dy/dx and the integral ∫ y dx.

Calculus Problem: Differentiation & Integration

Question:
If y = x5 + 3x2, find the derivative dy/dx and the integral ∫ y dx.

Solution:

Given Function:
y = x5 + 3x2

Step 1: Differentiation

Differentiating with respect to x:

dy/dx = (d/dx)(x5) + (d/dx)(3x2)

Using the power rule (d/dx)(xn) = nxn-1:

∴ dy/dx = 5x5-1 + 3(2x2-1)

= 5x4 + 3(2x)

Answer: dy/dx = 5x4 + 6x

Step 2: Integration

Now, integrating with respect to x:

∫ y dx = ∫ (x5 + 3x2) dx

Separating the terms:

= ∫ x5 dx + ∫ 3x2 dx

Using the integration rule ∫ xn dx = xn+1 / (n+1):

= x5+1 / (5+1) + 3 · x2+1 / (2+1)

= x6/6 + 3 · x3/3 + C

Here, the 3 cancels out.

Answer: ∫ y dx = x6/6 + x3 + C

*(Where C is the constant of integration)*

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