Evaluate the Integral

Problem

(∫_0^1)(x2+√(,x)*d(x))

Solution

1. Rewrite the integrand by expressing the square root as a power with a fractional exponent.

(∫_0^1)(x2+x(1/2)*d(x))

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

[(x3)/3+(x(3/2))/(3/2)]10

3. Simplify the coefficients of the integrated terms.

[1/3*x3+2/3*x(3/2)]10

4. Evaluate the expression at the upper limit of 1 and the lower limit of 0

(1/3*(1)3+2/3*(1)(3/2))−(1/3*(0)3+2/3*(0)(3/2))

5. Calculate the final numerical value by adding the fractions.

1/3+2/3=1

Final Answer

(∫_0^1)(x2+√(,x)*d(x))=1

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