## Integration as the Inverse of Differentiation

 3.1 Integration as the Inverse of Differentiation

Integration
 Integration is the reverse process of differentiation.

• A process is denoted by the symbol $$\int ... \,dx$$

Differentiation Integration
 $$\dfrac{d}{dx}[f(x)] = f'(x)$$
 $$\int f'(x) \ dx = f(x)$$

• In general,
 If $$\dfrac{d}{dx}[f(x)] = f'(x)$$, then the integral of $$f'(x)$$ with respct to $$x$$ is $$\int f'(x) \ dx = f(x)$$.

 Example Given $$\dfrac{d}{dx} (4x^2) = 8x$$, find $$\int 8x \ dx$$. Solution: Differentiation of $$4x^2$$ is $$8x$$. By the reverse of differentiation, the integration of $$8x$$ is $$4x^2$$. Hence, $$\int 8x \ dx = 4x^2$$.

## Integration as the Inverse of Differentiation

 3.1 Integration as the Inverse of Differentiation

Integration
 Integration is the reverse process of differentiation.

• A process is denoted by the symbol $$\int ... \,dx$$

Differentiation Integration
 $$\dfrac{d}{dx}[f(x)] = f'(x)$$
 $$\int f'(x) \ dx = f(x)$$

• In general,
 If $$\dfrac{d}{dx}[f(x)] = f'(x)$$, then the integral of $$f'(x)$$ with respct to $$x$$ is $$\int f'(x) \ dx = f(x)$$.

 Example Given $$\dfrac{d}{dx} (4x^2) = 8x$$, find $$\int 8x \ dx$$. Solution: Differentiation of $$4x^2$$ is $$8x$$. By the reverse of differentiation, the integration of $$8x$$ is $$4x^2$$. Hence, $$\int 8x \ dx = 4x^2$$.