Evaluate the Integral integral of 30x^(3/2) with respect to x

Problem

(∫_^)(30*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 30 outside the integral.

(∫_^)(30*x(3/2)*d(x))=30*(∫_^)(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. Calculate the new exponent by adding 1 to 3/2

3/2+1=5/2

5. Divide by the new exponent and add the constant of integration C

30⋅(x(5/2))/(5/2)+C

6. Simplify the expression by multiplying 30 by the reciprocal of 5/2 which is 2/5

30⋅2/5=12

Final Answer

(∫_^)(30*x(3/2)*d(x))=12*x(5/2)+C

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