Question 1

Evaluate the definite integral: ∫₀^(π/4) x·sec²(x) dx

Accepted Answer

We use Integration by Parts: ∫u dv = uv - ∫v du

Let u = x and dv = sec²(x)dx
Then du = dx and v = tan(x)

Applying Integration by Parts:
∫x·sec²(x) dx = x·tan(x) - ∫tan(x) dx

Now, ∫tan(x) dx = -ln|cos(x)| + C

So: ∫x·sec²(x) dx = x·tan(x) + ln|cos(x)| + C

Evaluating the definite integral from 0 to π/4:

At x = π/4:
(π/4)·tan(π/4) + ln|cos(π/4)| = (π/4)·1 + ln(1/√2) = π/4 - (1/2)ln(2)

At x = 0:
0·tan(0) + ln|cos(0)| = 0 + ln(1) = 0

Therefore:
∫₀^(π/4) x·sec²(x) dx = [π/4 - (1/2)ln(2)] - 0 = π/4 - (1/2)ln(2)

This can also be written as: (π - 2ln(2))/4 or π/4 - ln(√2)

Question 2

If y = x^x, then dy/dx is equal to:

Accepted Answer

Step 1: Take the natural logarithm of both sides. ln y = x · ln x. Step 2: Differentiate both sides with respect to x using implicit differentiation on the left and the product rule on the right. (1/y)(dy/dx) = ln x + x · (1/x) = ln x + 1. Step 3: Multiply both sides by y = x^x. dy/dx = x^x (ln x + 1). This uses the logarithmic differentiation technique, which is essential when the base is itself a variable.

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