Question

Make the appropriate trigonometric substitution to convert this integral, and write the integrated in the space provided.

Answer 1

Given

`\int (5)/((x^2+16)^2)dx`

Using the substitution x = 4 tan(t)

`\begin{gathered} \Rightarrow dx=4\sec ^2tdt \\ \Rightarrow\int (5)/((x^2+16)^2)dx=\int (5)/(((4\tan t)^2+16)^2)*4\sec ^2tdt \\ =\int (5)/((16\tan^2t+16)^2)*4\sec ^2tdt=\int (5)/(16^2(\tan^2t+1)^2)*4\sec ^2tdt \\ =\int (5)/(16^2(\tan^2t+1)^2)*4\sec ^2tdt=(5)/(64)\int (\sec^2t)/((\tan^2t+1)^2)dt \end{gathered}`

Since

`\begin{gathered} \tan ^2t+1=\sec ^2t \\ \Rightarrow(5)/(64)\int (\sec^2t)/((\tan^2t+1)^2)dt=(5)/(64)\int (\sec^2t)/((\sec^2t)^2)dt=(5)/(64)\int (1)/(\sec^2t)dt \end{gathered}`

(a)

`=(5)/(64)\int (1)/(\sec^2t)dt`

For b,

since we use the substitution

`\begin{gathered} x=4\tan t \\ \Rightarrow(x)/(4)=\tan t \\ \Rightarrow t=\tan ^(-1)((x)/(4)) \end{gathered}`

From the diagram above,

`\begin{gathered} \cos t=\frac{4}{\sqrt[]{x^2+16}} \\ \sin t=\frac{x}{\sqrt[]{x^2+16}} \\ \Rightarrow(5t)/(128)+(5\cos t\sin t)/(128)+C=(5)/(128)\tan ^(-1)((x)/(4))+(5)/(128)\frac{4}{\sqrt[]{x^2+16}}*\frac{x}{\sqrt[]{x^2+16}}+C \\ =(5)/(128)\tan ^(-1)((x)/(4))+(5x)/(32(x^2+16))+C \end{gathered}`

Hence, the answer in terms of x is

`I=(5)/(128)\tan ^(-1)((x)/(4))+(5x)/(32(x^2+16))+C`