**Note: This is the same thing you need to do when finding the y-intercept, C, for a linear function – see Functions: Linear functions.**

Standard integration

*Let f′(x) = 12x ^{2} − 2*

*Given that f (−1) = 1, find f (x).*

Separate summed parts (optional)

*∫ 12x ^{2} − 2dx = ∫ 12x^{2} dx + ∫ −2dx*

2. Integrate

*f(x) = ∫ 12x ^{2} dx + ∫ −2dx=*

^{12}/_{3} x^{3} − 2x + C

3. Fill in values of x and f(x) to find C

*Since f(−1) = 1,*

*4(−1) ^{3} −2(−1) + C = 1*

*C = 3 *

*So : f(x) = 4x ^{3} − 2x + 3*

### 5.1.1 Integration with an internal function

Example: Find the following integrals:

### 5.1.2 Integration by substitution

Integrate by substitution

Find

*∫ 3x*^{2}e^{x 3}dx- Identify the inside function u, this is the function whose derivative is also inside f(x)

*g(x) = u = x ^{3}*

2. Find the derivative u′ =

^{du}/_{dx}3. substitute u and ^{du}/_{dx } into the integral (this way dx cancels out)

4. Substitute u back to get a function with x

*∫ e ^{u} + C = e^{x3 } + C*