Evaluate the Integral integral of 6x^3 with respect to x

Problem

(∫_^)(6*x3*d(x))

Solution

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

2. Apply the constant multiple rule for integration by moving the constant 6 outside the integral.

(∫_^)(6*x3*d(x))=6*(∫_^)(x3*d(x))

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

6*(∫_^)(x3*d(x))=6⋅(x(3+1))/(3+1)+C

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

6⋅(x4)/4+C

5. Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2

(3*x4)/2+C

Final Answer

(∫_^)(6*x3*d(x))=(3*x4)/2+C

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