Home Calculus Integration Area Volume (ME1)

INTEGRATION

ℹ️ Anti-Derivatives

○ Differentiation starts with a function and produces it's derivative.

○ Integration starts with a derivative and produces the function (anti-derivative).

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ℹ️ HSC Reference Sheet Integration Rules

At the back of every HSC Mathematics exam is a Reference Sheet containing a list of useful formulas for each topic. The rules for integration include:

\(\int f'(x)[f(x)]^{n}dx = \frac{1}{n+1}[f(x)]^{n+1}+C \)

\( \int f'(x) \sin(f(x)) \,dx = -\cos(f(x))+C\)

\( \int f'(x) \cos(f(x)) \,dx = \sin(f(x))+C\)

\( \int f'(x) \sec^{2}(f(x)) \,dx = \tan(f(x))+C\)

\( \int f'(x) e^{f(x)} \,dx = e^{f(x)}+C\)

\( \int \frac{f'(x)}{f(x)} \,dx = \log_{e}{|f(x)|} + C \)

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🔄 Polynomial Powers

\(\int{x^{n}}dx = \frac{1}{n+1} \times x^{n+1}+C \)

Example: \( \int \left( x^{5} + \sqrt{x} + \frac{1}{x^{3}} \right) dx \)

Step 1: Write terms in index form: \( \int \left( x^{5} + \sqrt{x} + \frac{1}{x^{3}} \right) dx = \int \left( x^{5} + x^{\frac{1}{2}} + x^{-3} \right) dx \)

Step 2: Apply the rule:

\( \int \left( x^{5} + x^{\frac{1}{2}} + x^{-3} \right) dx = \frac{1}{5+1} x^{5+1} + \frac{1}{\frac{1}{2}+1}x^{\frac{1}{2}+1} + \frac{1}{-3+1}x^{-3+1} +C \)

Step 3: Don't forget to add 'C', the constant of integration. Rewrite into original form.

\( =\frac{1}{6} x^{6} + \frac{2}{3}x^{\frac{3}{2}} + \frac{1}{-2}x^{-2}+C =\frac{1}{6} x^{6} + \frac{2}{3}x \sqrt{x} - \frac{1}{2x^{2}}+C \)

Practice

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🔄 Reverse Chain Rule

\(\int f'(x)[f(x)]^{n}dx = \frac{1}{n+1} \times [f(x)]^{n+1}+C \)

Example: \( \int 5(3x-1)^{6} dx \)

Step 1: Identify f(x) and find f'(x): \( f(x)=3x-1 \Rightarrow f'(x)=3 \)

Step 2: Move unwanted constant multipliers across the integral sign:

\( \int 5(3x-1)^{6} dx = 5\int (3x-1)^{6} dx \)

Step 3: Multiply and divide the integrand by the constant f'(x):

\( 5\int (3x-1)^{6} dx = \frac{5}{3} \int 3(3x-1)^{6} dx \)

Step 5: Apply the rule to the integrand. Don't forget to add 'C'

\( \frac{5}{3} \int 3(3x-1)^{6} dx = \frac{5}{3} \times \frac{1}{7}(3x-1)^{7} + C = \frac{5}{21}(3x-1)^{7} + C\)

Practice

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🔄 cos Function

\( \int f'(x) \cos(f(x)) \,dx = \sin(f(x))+C\)

Example: \( \int (x + 1)\cos(x^{2} + 2x) \,dx \)

Step 1: Identify \( f(x) \): \( f(x) = x^{2} + 2x \)

Step 2: Find \( f'(x) \): is \( f'(x) = 2x + 2 = 2(x+1) \).

Step 3: Rewrite the integral into standard form. Multiply & divide integrand by the constant f'(x):

\( \int (x + 1)\cos(x^{2} + 2x) \,dx = \frac{1}{2} \int 2(x + 1)\cos(x^{2} + 2x)\,dx \)

Step 4: Apply the rule on the Reference Sheet and write down the antiderivative:

\( \int (x + 1)\cos(x^{2} + 2x) \,dx = \frac{1}{2} \sin(x^{2} + 2x) + C \)

Practice

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🔄 sin Function

\( \int f'(x) \sin(f(x)) \,dx = -\cos(f(x))+C\)

Example: \( \int \sin(3x + 2) \,dx \)

Step 1: Identify \( f(x) \): \( f(x) = 3x + 2 \)

Step 2: Find \( f'(x) \): is \( f'(x) = 3 \).

Step 3: Rewrite the integral into standard form: \( \int \sin(3x + 2) \,dx = \frac{1}{3} \int 3 \sin(3x + 2) \,dx \)

Step 4: Write down the antiderivative: \( \int \sin(3x + 2) \,dx = -\frac{1}{3} \cos(3x + 2) + C \)

Practice

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🔄 sec² Function

\( \int f'(x) \sec^{2}(f(x)) \,dx = \tan(f(x))+C\)

Example: \( \int 4 \sec^{2}(3x) \,dx \)

Step 1: Identify \( f(x) \): \( f(x) = 3x \)

Step 2: Find \( f'(x) \): is \( f'(x) = 3 \).

Step 3: Rewrite the integral into standard form: \( \int 4 \sec^{2}(3x) \,dx = \frac{4}{3} \int 3 \sec^{2}(3x) \,dx \)

Step 4: Write down the antiderivative: \( \int 4 \sec^{2}(3x) \,dx = \frac{4}{3} \tan(3x) + C \)

Practice

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🔄 Exponential Function

\( \int f'(x) e^{f(x)} \,dx = e^{f(x)}+C\)

\( \int f'(x) a^{f(x)} \,dx = \frac{a^{f(x)}}{\ln a}+C\)

Example: \( \int x e^{x^{2}} \,dx \)

Step 1: Identify \( f(x) \): \( f(x) = x^{2} \)

Step 2: Find \( f'(x) \): is \( f'(x) = 2x \).

Step 3: Rewrite the integral into standard form: \( \int x e^{x^{2}} \,dx = \frac{1}{2} \int 2x e^{x^{2}} \,dx \)

Step 4: Write down the antiderivative: \( \int x e^{x^{2}} \,dx = \frac{1}{2} e^{x^{2}} + C \)

Practice

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🔄 Rational Functions

\( \int \frac{f'(x)}{f(x)} \,dx = \ln {|f(x)|} + C \)

Example 5: \( \int \frac{5}{2x-1} \,dx \)

Step 1: Identify \( f(x) \): \( f(x) = 2x - 1 \)

Step 2: Find \( f'(x) \): is \( f'(x) = 2 \).

Step 3: Rewrite the integral into standard form: \( 5\int \frac{1}{2x-1} \,dx = \frac{5}{2} \int \frac{2}{2x-1} \,dx \)

Step 4: Write down the antiderivative: \( \int \frac{5}{2x-1} \,dx = \frac{5}{2} \ln|2x - 1| + C \)

Practice

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"The only way to learn mathematics is to do mathematics" - Paul R Halmos