|
|
| Line 3: |
Line 3: |
| | ::<math>\int \tan^4 x ~dx</math> | | ::<math>\int \tan^4 x ~dx</math> |
| | | | |
| | + | <hr> |
| | + | [[009B Sample Midterm 2, Problem 5 Solution|'''<u>Solution</u>''']] |
| | | | |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | [[009B Sample Midterm 2, Problem 5 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| − | !Foundations:
| |
| − | |-
| |
| − | |'''1.''' Recall the trig identity | |
| − | |-
| |
| − | | <math style="vertical-align: -1px">\sec^2x=\tan^2x+1</math>
| |
| − | |-
| |
| − | |'''2.''' Recall
| |
| − | |-
| |
| − | | <math style="vertical-align: -13px">\int \sec^2 x~dx=\tan x+C</math>
| |
| − | |-
| |
| − | |'''3.''' How would you integrate <math style="vertical-align: -12px">\int \sec^2(x)\tan(x)~dx?</math>
| |
| − | |-
| |
| − | |
| |
| − | You can use <math style="vertical-align: 0px">u</math>-substitution.
| |
| − | |-
| |
| − | | Let <math style="vertical-align: -2px">u=\tan x.</math>
| |
| − | |-
| |
| − | | Then, <math style="vertical-align: -5px">du=\sec^2(x)dx.</math>
| |
| − | |-
| |
| − | |
| |
| − | Thus,
| |
| − | |-
| |
| − | | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\int \sec^2(x)\tan(x)~dx} & = & \displaystyle{\int u~du}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{u^2}{2}+C}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{\tan^2x}{2}+C.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| | | | |
| | | | |
| − | '''Solution:'''
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |First, we write
| |
| − | |-
| |
| − | | <math style="vertical-align: -13px">\int \tan^4(x)~dx=\int \tan^2(x) \tan^2(x)~dx.</math>
| |
| − | |-
| |
| − | |Using the trig identity <math style="vertical-align: -5px">\sec^2(x)=\tan^2(x)+1,</math>
| |
| − | |-
| |
| − | |we have
| |
| − | |-
| |
| − | | <math style="vertical-align: -5px">\tan^2(x)=\sec^2(x)-1.</math>
| |
| − | |-
| |
| − | |Plugging in the last identity into one of the <math style="vertical-align: -5px">\tan^2(x),</math> we get
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\int \tan^4(x)~dx} & = & \displaystyle{\int \tan^2(x) (\sec^2(x)-1)~dx}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\int \tan^2(x)\sec^2(x)~dx-\int \tan^2(x)~dx}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\int \tan^2(x)\sec^2(x)~dx-\int (\sec^2x-1)~dx}
| |
| − | \end{array}</math>
| |
| − | |-
| |
| − | |by using the identity again on the last equality.
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |So, we have
| |
| − | |-
| |
| − | | <math style="vertical-align: -13px">\int \tan^4(x)~dx=\int \tan^2(x)\sec^2(x)~dx-\int (\sec^2x-1)~dx.</math>
| |
| − | |-
| |
| − | |For the first integral, we need to use <math style="vertical-align: 0px">u</math>-substitution.
| |
| − | |-
| |
| − | |Let <math style="vertical-align: -5px">u=\tan(x).</math>
| |
| − | |-
| |
| − | |Then, <math style="vertical-align: -5px">du=\sec^2(x)dx.</math>
| |
| − | |-
| |
| − | |So, we have
| |
| − | |-
| |
| − | | <math style="vertical-align: -13px">\int \tan^4(x)~dx=\int u^2~du-\int (\sec^2(x)-1)~dx.</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 3:
| |
| − | |-
| |
| − | |We integrate to get
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\int \tan^4(x)~dx} & = & \displaystyle{\frac{u^3}{3}-(\tan(x)-x)+C}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{\tan^3(x)}{3}-\tan(x)+x+C.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Final Answer:
| |
| − | |-
| |
| − | | <math>\frac{\tan^3(x)}{3}-\tan(x)+x+C</math>
| |
| − | |}
| |
| | [[009B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | | [[009B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] |
