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Integration I

Finding functions by integrating

Finding functions by integrating

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Integration I

Integrating x^n

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Finding functions by integrating

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Differentiation I

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Explainer Video

Tutor: Bilal

Summary

Finding functions by integrating

In a nutshell

When given the gradient function f′(x)f'(x)f′(x)​, it is possible to find the function f(x)f(x)f(x) by integrating. However, this gives f(x)f(x)f(x) up to an unknown constant. This constant can be found if a specific point of f(x)f(x)f(x)​ is known.



Finding functions

To find a function y=f(x)y=f(x)y=f(x)​ given a gradient function dydx=f′(x)\dfrac{dy}{dx}=f'(x)dxdy​=f′(x) and a point P(a,b)P(a,b)P(a,b), follow this procedure.


procedure

​1.

Integrate the gradient function to obtain y=f(x)+Cy=f(x)+Cy=f(x)+C.

2.

Find the value of CCC​ by substituting in the point (a,b)(a,b)(a,b). So when x=a,y=bx=a,y=bx=a,y=b​.

3.

Rewrite the function with the known value of CCC​.


Example 1

A function y=f(x)y=f(x)y=f(x) has gradient function dydx=6x+12x\dfrac{dy}{dx}=\dfrac{6}{\sqrt{x}}+12xdxdy​=x​6​+12x. The function passes through the point P(1,5)P(1,5)P(1,5). Find the function f(x)f(x)f(x).


Integrate the gradient function to find f(x)f(x)f(x) up to a constant:

f(x)=∫dydx dx=∫(6x+12x) dx=∫6x−12 dx+∫12x1 dx+C=612x12+122x2+C=12x12+6x2+C\begin{aligned}f(x)&=\int \dfrac{dy}{dx}\, dx\\&=\int (\dfrac{6}{\sqrt{x}}+12x) \, dx \\&=\int 6x^{-\frac{1}{2}}\, dx + \int 12x^1\, dx +C\\&=\dfrac{6}{\frac{1}{2}}x^\frac12+\dfrac{12}{2}x^2 +C \\&=12x^{\frac12}+6x^2+C\end{aligned}f(x)​=∫dxdy​dx=∫(x​6​+12x)dx=∫6x−21​dx+∫12x1dx+C=21​6​x21​+212​x2+C=12x21​+6x2+C​​​


Therefore, y=12x12+6x2+Cy=12x^{\frac12}+6x^2+Cy=12x21​+6x2+C.


Use the point PPP to find the value of the constant CCC:

x=1,y=5y=12x12+6x2+C5=12(1)12+6(1)2+C5=12+6+C5=18+C−13=Cx=1,y=5\\ \begin{aligned}y&=12x^{\frac12}+6x^2+C\\5&=12(1)^{\frac12}+6(1)^2+C\\5&=12+6+C\\5&=18+C\\-13&=C\end{aligned}x=1,y=5y555−13​=12x21​+6x2+C=12(1)21​+6(1)2+C=12+6+C=18+C=C​​​


Rewrite the function with the known value of CCC:

​f(x)=12x12+6x2−13‾\underline{f(x)=12x^{\frac12}+6x^2-13}f(x)=12x21​+6x2−13​​​


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FAQs - Frequently Asked Questions

How do you find the constant of integration?

The constant of integration, or '+c', can be found by substituting a known pair of coordinates into the function f(x).

How many functions are there that have the same gradient function?

There are infinitely many functions that have the same gradient function, as you can choose infinitely many different values of C, the constant of integration.

How do you find a function given the gradient function and a point?

Integrate the gradient function and use the point to find the value of the constant of integration.

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