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Question 1 of 6
Find the integral
∫ sin8x dx
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Integrals of Trigonometric Functions
Integrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c
Substitute the components into the formula
∫f(g(x))dx |
= |
f(g(x))⋅1g′(x)+c |
|
∫sin(8x)dx |
= |
−cos8x⋅1g′(8x)+c |
Substitute known values |
|
|
= |
−cos8x⋅18+c |
Evaluate |
|
|
= |
−18cos8x+c |
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Question 2 of 6
Find the integral
∫ 4sec22x dx
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Integrals of Trigonometric Functions
Integrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c
Substitute the components into the formula
∫f(g(x))dx |
= |
f(g(x))⋅1g′(x)+c |
|
∫sec2(2x)dx |
= |
tan2x⋅1g′(2x)+c |
Substitute known values |
|
|
= |
tan2x⋅12+c |
Evaluate |
|
|
= |
2tan2x+c |
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Question 3 of 6
Find the integral
∫ sin(π-x) dx
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Integrals of Trigonometric Functions
Integrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c
Substitute the components into the formula
∫f(g(x))dx |
= |
f(g(x))⋅1g′(x)+c |
|
∫sin(π−x)dx |
= |
−cos(π−x)⋅1g′(π−x)+c |
Substitute known values |
|
|
= |
−cos(π−x)⋅1−1+c |
Evaluate |
|
|
= |
cos(π−x)+c |
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Question 4 of 6
Find the integral
∫ [sec22x-cos x2] dx
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Integrals of Trigonometric Functions
Integrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c
Substitute the components of each term into the formula
First term
∫f(g(x))dx |
= |
f(g(x))⋅1g′(x)+c |
|
∫sec22xdx |
= |
tan2x⋅1g′(2x) |
Substitute known values |
|
|
= |
tan2x⋅12 |
Evaluate |
|
|
= |
12tan2x |
Second term
∫f(g(x))dx |
= |
f(g(x))⋅1g′(x)+c |
|
∫−cos(x2)dx |
= |
−sinx2⋅1g′(x2) |
Substitute known values |
|
|
= |
−cosx2⋅2 |
Evaluate |
|
|
= |
−2cosx2 |
Finally, combine the two terms and add the constant
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Question 5 of 6
Find the integral
∫ tan2x dx
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Integrals of Trigonometric Functions
Integrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c
First, convert the function into a derivable function
Take note that sec2=1+tan2x
sec2 |
= |
1+tan2x |
sec2 -1 |
= |
1+tan2x -1 |
Subtract 1 from both sides |
sec2-1 |
= |
tan2x |
Therefore, we can use sec2-1 as a derivable substitute
Finally, substitute the components into the formula
∫f(g(x))dx |
= |
f(g(x))⋅1g′(x)+c |
|
∫sec2x−1dx |
= |
tanx−x+c |
Substitute known values and integrate |
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Question 6 of 6
Find the integral
∫ tanx dx
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Integrals of Trigonometric Functions
Integrating Trigonometric Functions
∫f(g(x))dx=f(g(x))⋅1g′(x)+c
First, convert the function into a derivable function
Take note that tan x=sin xcos x
Therefore, we can use sin xcos x as a derivable substitute
Next, take note that ddxcos x=-sin x
This means that the function satisfies the derivative of a natural logarithm f′(x)f(x), if the equation is balanced
We can use -1 as a constant to balance the function
Finally, integrate the function into a natural logarithm
-∫ -sin xcos x |
= |
-ln (cos x)+c |