Find the Integral 12x(x^2+3)^5

Problem

(∫_^)(12*x*(x2+3)5*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x2+3 is a multiple of the outer factor 12*x

2. Define the variable u and its derivative.

u=x2+3

d(u)/d(x)=2*x

d(u)=2*x*d(x)

3. Rewrite the original integral in terms of u by factoring the constant.

(∫_^)(12*x*(x2+3)5*d(x))=(∫_^)(6*(x2+3)5*(2*x)*d(x))

(∫_^)(6*u5*d(u))

4. Integrate with respect to u using the power rule.

6⋅(u6)/6+C

u6+C

5. Substitute the original expression x2+3 back in for u

(x2+3)6+C

Final Answer

(∫_^)(12*x*(x2+3)5*d(x))=(x2+3)6+C

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