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Find $$\int \frac{1}{(5-4x)^2} \,dx, \quad x < \frac{5}{4}$$ - Scottish Highers Maths - Question 13 - 2017

Question 13

$Find-$$\int-\frac{1}{(5-4x)^2}-\,dx,-\quad-x-<-\frac{5}{4}$$-Scottish Highers Maths-Question 13-2017.png$

Find $$\int \frac{1}{(5-4x)^2} \,dx, \quad x < \frac{5}{4}$$

Worked Solution & Example Answer:Find $$\int \frac{1}{(5-4x)^2} \,dx, \quad x < \frac{5}{4}$$ - Scottish Highers Maths - Question 13 - 2017

Step 1

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Answer

We start with the integral expressed as:

$\int \frac{1}{(5 - 4x)^2} \,dx$

Step 2

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Answer

To integrate this expression, we can use the substitution method. Let:

\Rightarrow \ dx = -\frac{1}{4} \,du$$

Step 3

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Answer

Substituting for $u$ and $dx$ , the integral becomes:

$\int \frac{1}{u^2} \left(-\frac{1}{4} \,du\right) = -\frac{1}{4} \int u^{-2} \,du$

Step 4

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Answer

Integrating $u^{-2}$ gives:

$-\frac{1}{4} \left(-u^{-1}\right) + c = \frac{1}{4(5 - 4x)} + c$

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