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Indefinite Integrals of Trig FunctionsIndefinite Integrals of Trig Functions
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Question 1 of 6
1. Question
Find the integral∫ sin8x dx∫ sin8x dx- 1.
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Integrals of Trigonometric Functions
∫ cos=sin∫ cos=sin∫ sin=-cos∫ sin=−cos∫ sec2=tan∫ sec2=tanIntegrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c∫f(g(x))dx=f(g(x))⋅1g′(x)+cSubstitute the components into the formula∫f(g(x))dx∫f(g(x))dx == f(g(x))⋅1g′(x)+cf(g(x))⋅1g′(x)+c ∫sin(8x)dx∫sin(8x)dx == −cos8x⋅1g′(8x)+c−cos8x⋅1g′(8x)+c Substitute known values == −cos8x⋅18+c−cos8x⋅18+c Evaluate == −18cos8x+c−18cos8x+c -18cos 8x+c−18cos 8x+c -
Question 2 of 6
2. Question
Find the integral∫ 4sec22x dx∫ 4sec22x dx-
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Chapters- Chapters
Integrals of Trigonometric Functions
∫ cos=sin∫ cos=sin∫ sin=-cos∫ sin=−cos∫ sec2=tan∫ sec2=tanIntegrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c∫f(g(x))dx=f(g(x))⋅1g′(x)+cSubstitute the components into the formula∫f(g(x))dx∫f(g(x))dx == f(g(x))⋅1g′(x)+cf(g(x))⋅1g′(x)+c ∫sec2(2x)dx∫sec2(2x)dx == tan2x⋅1g′(2x)+ctan2x⋅1g′(2x)+c Substitute known values == tan2x⋅12+ctan2x⋅12+c Evaluate == 2tan2x+c2tan2x+c 2 tan 2x+c2 tan 2x+c -
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Question 3 of 6
3. Question
Find the integral∫ sin(π-x) dx∫ sin(π−x) dx-
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Need TextPlayCurrent Time 0:00/Duration Time 0:00Remaining Time -0:00Stream TypeLIVELoaded: 0%Progress: 0%0:00Fullscreen00:00MutePlayback Rate1x- 2x
- 1.5x
- 1.25x
- 1x
- 0.75x
- 0.5x
Subtitles- subtitles off
Captions- captions off
- English
Chapters- Chapters
Integrals of Trigonometric Functions
∫ cos=sin∫ cos=sin∫ sin=-cos∫ sin=−cos∫ sec2=tan∫ sec2=tanIntegrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c∫f(g(x))dx=f(g(x))⋅1g′(x)+cSubstitute the components into the formula∫f(g(x))dx∫f(g(x))dx == f(g(x))⋅1g′(x)+cf(g(x))⋅1g′(x)+c ∫sin(π−x)dx∫sin(π−x)dx == −cos(π−x)⋅1g′(π−x)+c−cos(π−x)⋅1g′(π−x)+c Substitute known values == −cos(π−x)⋅1−1+c−cos(π−x)⋅1−1+c Evaluate == cos(π−x)+ccos(π−x)+c cos (π-x)+ccos (π−x)+c -
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Question 4 of 6
4. Question
Find the integral∫ [sec22x-cos x2] dx∫ [sec22x−cos x2] dx-
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Need TextPlayCurrent Time 0:00/Duration Time 0:00Remaining Time -0:00Stream TypeLIVELoaded: 0%Progress: 0%0:00Fullscreen00:00MutePlayback Rate1x- 2x
- 1.5x
- 1.25x
- 1x
- 0.75x
- 0.5x
Subtitles- subtitles off
Captions- captions off
- English
Chapters- Chapters
Integrals of Trigonometric Functions
∫ cos=sin∫ cos=sin∫ sin=-cos∫ sin=−cos∫ sec2=tan∫ sec2=tanIntegrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c∫f(g(x))dx=f(g(x))⋅1g′(x)+cSubstitute the components of each term into the formulaFirst term∫f(g(x))dx∫f(g(x))dx == f(g(x))⋅1g′(x)+cf(g(x))⋅1g′(x)+c ∫sec22xdx∫sec22xdx == tan2x⋅1g′(2x)tan2x⋅1g′(2x) Substitute known values == tan2x⋅12tan2x⋅12 Evaluate == 12tan2x12tan2x Second term∫f(g(x))dx∫f(g(x))dx == f(g(x))⋅1g′(x)+cf(g(x))⋅1g′(x)+c ∫−cos(x2)dx∫−cos(x2)dx == −sinx2⋅1g′(x2)−sinx2⋅1g′(x2) Substitute known values == −cosx2⋅2−cosx2⋅2 Evaluate == −2cosx2−2cosx2 Finally, combine the two terms and add the constant12tan2x−2sinx2+c12tan2x−2sinx2+c 12 tan 2x-2 sin x2+c12 tan 2x−2 sin x2+c -
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Question 5 of 6
5. Question
Find the integral∫ tan2x dx∫ tan2x dx-
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Excellent!
Incorrect
Need TextPlayCurrent Time 0:00/Duration Time 0:00Remaining Time -0:00Stream TypeLIVELoaded: 0%Progress: 0%0:00Fullscreen00:00MutePlayback Rate1x- 2x
- 1.5x
- 1.25x
- 1x
- 0.75x
- 0.5x
Subtitles- subtitles off
Captions- captions off
- English
Chapters- Chapters
Integrals of Trigonometric Functions
∫ cos=sin∫ cos=sin∫ sin=-cos∫ sin=−cos∫ sec2=tan∫ sec2=tanIntegrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c∫f(g(x))dx=f(g(x))⋅1g′(x)+cFirst, convert the function into a derivable functionTake note that sec2=1+tan2xsec2=1+tan2xsec2sec2 == 1+tan2x1+tan2x sec2sec2 -1−1 == 1+tan2x1+tan2x -1−1 Subtract 11 from both sides sec2-1sec2−1 == tan2xtan2x Therefore, we can use sec2-1sec2−1 as a derivable substituteFinally, substitute the components into the formula∫f(g(x))dx∫f(g(x))dx == f(g(x))⋅1g′(x)+cf(g(x))⋅1g′(x)+c ∫sec2x−1dx∫sec2x−1dx == tanx−x+ctanx−x+c Substitute known values and integrate tan x-x+ctan x−x+c -
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Question 6 of 6
6. Question
Find the integral∫ tanx dx∫ tanx dx-
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Hint
Help VideoCorrect
Keep Going!
Incorrect
Need TextPlayCurrent Time 0:00/Duration Time 0:00Remaining Time -0:00Stream TypeLIVELoaded: 0%Progress: 0%0:00Fullscreen00:00MutePlayback Rate1x- 2x
- 1.5x
- 1.25x
- 1x
- 0.75x
- 0.5x
Subtitles- subtitles off
Captions- captions off
- English
Chapters- Chapters
Integrals of Trigonometric Functions
∫ cos=sin∫ cos=sin∫ sin=-cos∫ sin=−cos∫ sec2=tan∫ sec2=tanIntegrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c∫f(g(x))dx=f(g(x))⋅1g′(x)+cFirst, convert the function into a derivable functionTake note that tan x=sin xcos xtan x=sin xcos xTherefore, we can use sin xcos xsin xcos x as a derivable substituteNext, take note that ddxcos x=-sin xddxcos x=−sin xThis means that the function satisfies the derivative of a natural logarithm f′(x)f(x), if the equation is balancedWe can use -1 as a constant to balance the function= -∫ -sin xcos x Finally, integrate the function into a natural logarithm-∫ -sin xcos x = -ln (cos x)+c -ln (cos x)+c -
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Quizzes
- Indefinite Integrals 1
- Indefinite Integrals 2
- Indefinite Integrals 3
- Indefinite Integrals of Exponential Functions
- Indefinite Integrals of Logarithmic Functions 1
- Indefinite Integrals of Logarithmic Functions 2
- Indefinite Integrals of Trig Functions
- Definite Integrals
- Definite Integrals of Exponential Functions
- Definite Integrals of Logarithmic Functions
- Definite Integrals of Trig Functions
- Areas Between Curves and the Axis 1
- Areas Between Curves and the Axis 2
- Area Between Curves
- Volumes of Revolution 1
- Volumes of Revolution 2
- Volumes of Revolution 3
- Trapezoidal Rule
- Simpsons Rule
- Applications of Integration for Trig Functions