Evaluate the Integral integral of 15x^(3/2) with respect to x

Problem

(∫_^)(15*x(3/2)*d(x))

Solution

1. Identify the constant and the power of x in the integrand.

2. Apply the constant multiple rule by moving the constant 15 outside the integral.

(∫_^)(15*x(3/2)*d(x))=15*(∫_^)(x(3/2)*d(x))

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

4. Add 1 to the exponent: 3/2+1=5/2

5. Divide by the new exponent: (x(5/2))/(5/2)

15*(∫_^)(x(3/2)*d(x))=15⋅(x(5/2))/(5/2)+C

6. Simplify the expression by multiplying 15 by the reciprocal of the fraction 5/2 which is 2/5

15⋅2/5=6

7. Combine the results to find the final antiderivative.

Final Answer

(∫_^)(15*x(3/2)*d(x))=6*x(5/2)+C

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