Evaluate the Integral

Problem

(∫_0^1)(√(,x)*d(x))

Solution

1. Rewrite the integrand using a fractional exponent to prepare for the power rule.

√(,x)=x(1/2)

2. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1) for n≠−1

(∫_^)(x(1/2)*d(x))=(x(1/2+1))/(1/2+1)

3. Simplify the resulting expression and the exponent.

(x(3/2))/(3/2)=2/3*x(3/2)

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 1 and the lower limit 0

[2/3*x(3/2)]10

5. Substitute the limits into the expression.

2/3*(1)(3/2)−2/3*(0)(3/2)

6. Calculate the final numerical value.

2/3*(1)−2/3*(0)=2/3

Final Answer

(∫_0^1)(√(,x)*d(x))=2/3

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