Evaluate the Integral integral of 8x with respect to x

Problem

(∫_^)(8*x*d(x))

Solution

1. Identify the constant and the variable part of the integrand. The constant is 8 and the variable part is x

2. Apply the constant multiple rule for integration by moving the 8 outside the integral sign.

(∫_^)(8*x*d(x))=8*(∫_^)(x*d(x))

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

8*(∫_^)(x1*d(x))=8⋅(x(1+1))/(1+1)

4. Simplify the expression by performing the arithmetic in the exponent and the denominator.

8⋅(x2)/2

5. Reduce the fraction by dividing 8 by 2

4*x2

6. Add the constant of integration C to represent the family of antiderivatives.

4*x2+C

Final Answer

(∫_^)(8*x*d(x))=4*x2+C

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