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a) What is the inside function? What is its derivative?b) Perform the integration using u-substitution.

a) What is the inside function? What is its derivative?b) Perform the integration-example-1
User NickZeng
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1 Answer

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Given the integral function below


\int(-6x)/((4x^2-9)^4)dx

a) The inside function is


\begin{gathered} 4x^2-9 \\ \end{gathered}
\therefore u=4x^2-9

The derivative of the inside function is


\begin{gathered} u^(\prime)=8x \\ (du)/(dx)=8x \\ dx=(du)/(8x) \end{gathered}

b) Therefore, the integral can be written as:


\int(-6x)/((4x^2-9)^4)dx\rightarrow\int(-6x)/(u^4)*(du)/(8x)
\rightarrow\int(-3)/(4u^4)du\rightarrow-(3)/(4)\int u^(-4)du
\begin{gathered} \rightarrow-(3)/(4)((u^(-4+1))/(-4+1))+C \\ \rightarrow-(3)/(4)*(u^(-3))/(-3)+C \\ \rightarrow(1)/(4)*(1)/(u^3)+C \end{gathered}
\begin{gathered} (1)/(4u^3)+C \\ Recall\text{ u=}4x^2-9 \\ \therefore \\ \int(-6x)/((4x^2-9)^4)dx=(1)/(4(4x^2-9)^3)+C \end{gathered}

Final answer:

a) The inside function is 4x² - 9, and the derivative is 8x

b) The integration by substitution is


\begin{equation*} (1)/(4(4x^2-9)^3)+C \end{equation*}

User Nicko Glayre
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