Find the Function f''(x)=12x+sin(x)

Problem

ƒ(x)″=12*x+sin(x)

Solution

1. Integrate the second derivative ƒ(x)″ with respect to x to find the first derivative ƒ(x)′

(∫_^)((12*x+sin(x))*d(x))

2. Apply the power rule for integration (∫_^)(xn*d(x))=(x(n+1))/(n+1) and the trigonometric integral (∫_^)(sin(x)*d(x))=−cos(x)

ƒ(x)′=(12*x2)/2−cos(x)+(C_1)

3. Simplify the expression for the first derivative.

ƒ(x)′=6*x2−cos(x)+(C_1)

4. Integrate the first derivative ƒ(x)′ with respect to x to find the general function ƒ(x)

(∫_^)((6*x2−cos(x)+(C_1))*d(x))

5. Apply the integration rules again, noting that (∫_^)(cos(x)*d(x))=sin(x) and the integral of a constant (C_1) is (C_1)*x

ƒ(x)=(6*x3)/3−sin(x)+(C_1)*x+(C_2)

6. Simplify the final expression for the function.

ƒ(x)=2*x3−sin(x)+(C_1)*x+(C_2)

Final Answer

ƒ(x)=2*x3−sin(x)+(C_1)*x+(C_2)

Want more problems? Check here!