Evaluate the Integral

Problem

(∫_^)(√(,x)*d(x))

Solution

1. Rewrite the radical expression using a fractional exponent to make it easier to apply the power rule for integration.

√(,x)=x^(1/2)

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

(∫_^)(x^(1/2)*d(x))=(x^(1/2+1))/(1/2+1)+C

3. Simplify the exponent and the denominator by performing the addition.

(x^(3/2))/(3/2)+C

4. Multiply by the reciprocal of the denominator to find the final coefficient.

2/3*x^(3/2)+C

Final Answer

(∫_^)(√(,x)*d(x))=2/3*x^(3/2)+C

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