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SS3: MATHEMATICS - 2ND TERM

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  1. Matrices I | Week 1
    6 Topics
  2. Matrices II | Week 2
    1 Topic
    |
    1 Quiz
  3. Commercial Arithmetic | Week 3
    7 Topics
    |
    1 Quiz
  4. Coordinate Geometry | Week 4
    8 Topics
    |
    1 Quiz
  5. Differentiation of Algebraic Expressions | Week 5 & 6
    7 Topics
  6. Application of Differentiation | Week 7
    4 Topics
    |
    1 Quiz
  7. Integration | Week 8
    8 Topics
    |
    1 Quiz



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Lesson 7, Topic 7
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Integration of Products (Integration by Parts)

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We often need to integrate a product where either function is not the derivative of the other.

If u and v are functions of x, then

\( \frac{d(uv)}{dx} = \scriptsize u \normalsize\frac{dv}{dx} + \scriptsize v \normalsize\frac{du}{dx} \)

Integrating both sides with respect to x

uv = \(\int \scriptsize u \normalsize\frac{dv}{dx} \scriptsize dx + \int \scriptsize v \normalsize\frac{du}{dx} \scriptsize dx \)

uv = \(\int \scriptsize udv + \normalsize \int \scriptsize vdu \)

\(\int \scriptsize udv = uv\; – \normalsize \int \scriptsize vdu \)

Guides for selecting u and dv in a given function. 

Select u in this order below:

L: logarithmic function

I: inverse trigonometric function

A: algebraic function

T: trigonometric function

E: exponential function

Example:

Evaluate \(\scriptsize \int x^3 \ln x dx \)

Solution: \(\int \scriptsize udv = uv\; – \normalsize \int \scriptsize vdu \)

Let u = ln x since it’s a logarithmic function, then dv = \( \scriptsize x^3 dx \)

We obtain \( \frac{du}{dx} = \frac{1}{dx}, \scriptsize du = \normalsize \frac{1}{x} \scriptsize dx\) and

v = \(\scriptsize \int x^3 dx = \normalsize \frac{x^4}{4} \)

Substituting into the formula,

\(\int \scriptsize udv = uv\; – \normalsize \int \scriptsize vdu \) we have

= \( \scriptsize \ln x \normalsize \frac{x^4}{4} \; – \frac{x^4}{4} \frac{1}{x} \scriptsize dx \)

= \( \scriptsize \ln x \normalsize \frac{x^4}{4} \; – \frac{1}{4}\scriptsize \int x^3 dx \)

= \( \scriptsize \ln x \normalsize \frac{x^4}{4} \; – \frac{1}{4}\left( \frac{x^4}{4} \right) \scriptsize + c \)

\(\scriptsize \int x^3 \ln x dx = \normalsize \frac{x^4}{4} \left ( \scriptsize \ln x \; – \normalsize \frac{1}{4} \right)\scriptsize + c \)

Exercise

Evaluate the following

a. \(\scriptsize \int x^2 \ln x dx \)

b. \(\scriptsize \int x^2 e^{3x} dx \)

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