Find the function f(x), whose derivative f'(x)=12x^{11}

Question:

Find the function {eq}f(x) {/eq}, whose derivative {eq}f'(x)=12x^{11} {/eq}

The Function using Anti-derivative:

The anti-derivative is the reverse process to find the function of the derivative. For example, {eq}\displaystyle \int f'(x) = f(x). {/eq}

Some of the various application of anti-derivatives:

1. Move the constant out: {eq}\displaystyle \int a\cdot f\left(v\right)dv=a\cdot \int f\left(v\right)dv. {/eq}

2. The power rule: {eq}\displaystyle \int v^adv=\frac{v^{a+1}}{a+1}, \quad a\ne -1. {/eq}

Answer and Explanation:

Solving for $$\displaystyle f'(x)=12x^{11} $$

Take the integration both sides.

$$\displaystyle \int f'(x) = \int 12x^{11} dx $$

Move the constant out.

$$\displaystyle f(x) = 12\cdot \int x^{11}dx $$

Apply the power rule.

$$\displaystyle f(x) = 12\cdot \frac{x^{11+1}}{11+1}+C $$

Simplify:

$$\displaystyle f(x) = x^{12}+C. $$

Where C is constant of the integration.


Learn more about this topic:

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Antiderivative: Rules, Formula & Examples

from Calculus: Help and Review

Chapter 8 / Lesson 12
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