Years
>
Year 12>
Integration>
Integral of a Trigonometric Function>
Integral of a Trigonometric Function 1Integral of a Trigonometric Function 1
Try VividMath Premium to unlock full access
Time limit: 0
Quiz summary
0 of 4 questions completed
Questions:
- 1
- 2
- 3
- 4
Information
–
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Loading...
- 1
- 2
- 3
- 4
- Answered
- Review
-
Question 1 of 4
1. Question
Find the integral`int sin8x dx`Hint
Help VideoCorrect
Well Done!
Incorrect
Integrals of Trigonometric Functions
`int \text(cos)=\text(sin)``int \text(sin)=-\text(cos)``int \text(sec)^2=\text(tan)`Integrating Trigonometric Functions
$$\int f(\color{#004ec4}{g(x)}) dx=f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$Substitute the components into the formula$$\int f(\color{#004ec4}{g(x)}) dx$$ `=` $$f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$ $$\int \text{sin}(\color{#004ec4}{8x}) dx$$ `=` $$-\text{cos}\;8x\cdot\frac{1}{\color{#004ec4}{g'(8x)}} +c$$ Substitute known values `=` $$-\text{cos}\;8x\cdot\frac{1}{8} +c$$ Evaluate `=` $$-\frac{1}{8}\;\text{cos}\;8x +c$$ `-1/8 \text(cos) 8x+c` -
Question 2 of 4
2. Question
Find the integral`int 4sec^2 2x dx`Hint
Help VideoCorrect
Great Work!
Incorrect
Integrals of Trigonometric Functions
`int \text(cos)=\text(sin)``int \text(sin)=-\text(cos)``int \text(sec)^2=\text(tan)`Integrating Trigonometric Functions
$$\int f(\color{#004ec4}{g(x)}) dx=f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$Substitute the components into the formula$$\int f(\color{#004ec4}{g(x)}) dx$$ `=` $$f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$ $$\int \text{sec}^2(\color{#004ec4}{2x}) dx$$ `=` $$\text{tan}\;2x\cdot\frac{1}{\color{#004ec4}{g'(2x)}} +c$$ Substitute known values `=` $$\text{tan}\;2x\cdot\frac{1}{2} +c$$ Evaluate `=` $$2\;\text{tan}\;2x +c$$ `2 \text(tan) 2x+c` -
Question 3 of 4
3. Question
Find the integral`int sin(pi-x) dx`Hint
Help VideoCorrect
Nice Job!
Incorrect
Integrals of Trigonometric Functions
`int \text(cos)=\text(sin)``int \text(sin)=-\text(cos)``int \text(sec)^2=\text(tan)`Integrating Trigonometric Functions
$$\int f(\color{#004ec4}{g(x)}) dx=f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$Substitute the components into the formula$$\int f(\color{#004ec4}{g(x)}) dx$$ `=` $$f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$ $$\int \text{sin}(\color{#004ec4}{\pi-x}) dx$$ `=` $$-\text{cos}\;(\pi-x)\cdot\frac{1}{\color{#004ec4}{g'(\pi-x)}} +c$$ Substitute known values `=` $$-\text{cos}\;(\pi-x)\cdot\frac{1}{-1} +c$$ Evaluate `=` $$\text{cos}\;(\pi-x) +c$$ `\text(cos) (\pi-x)+c` -
Question 4 of 4
4. Question
Find the integral`int [sec^2 2x-cos x/2] dx`Hint
Help VideoCorrect
Correct!
Incorrect
Integrals of Trigonometric Functions
`int \text(cos)=\text(sin)``int \text(sin)=-\text(cos)``int \text(sec)^2=\text(tan)`Integrating Trigonometric Functions
$$\int f(\color{#004ec4}{g(x)}) dx=f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$Substitute the components of each term into the formulaFirst term$$\int f(\color{#004ec4}{g(x)}) dx$$ `=` $$f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$ $$\int \text{sec}^2\color{#004ec4}{2x}\;dx$$ `=` $$\text{tan}\;2x\cdot\frac{1}{\color{#004ec4}{g'(2x)}}$$ Substitute known values `=` $$\text{tan}\;2x\cdot\frac{1}{2}$$ Evaluate `=` $$\frac{1}{2}\text{tan}\;2x$$ Second term$$\int f(\color{#004ec4}{g(x)}) dx$$ `=` $$f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$$ $$\int -\text{cos}\left(\color{#004ec4}{\frac{x}{2}}\right)\;dx$$ `=` $$-\text{sin}\;\frac{x}{2}\cdot\frac{1}{\color{#004ec4}{g'(\frac{x}{2})}}$$ Substitute known values `=` $$-\text{cos}\;\frac{x}{2}\cdot2$$ Evaluate `=` $$-2\text{cos}\;\frac{x}{2}$$ Finally, combine the two terms and add the constant$$\frac{1}{2}\text{tan}\;2x-2\text{sin}\;\frac{x}{2} +c$$ `1/2 \text(tan) 2x-2 \text(sin) x/2+c`
Quizzes
- Indefinite Integrals 1
- Indefinite Integrals 2
- Indefinite Integrals 3
- Definite Integrals
- Areas Between Curves and the Axis 1
- Areas Between Curves and the Axis 2
- The Area Between Curves
- Volumes of Revolution 1
- Volumes of Revolution 2
- Volumes of Revolution 3
- Trapezoidal Rule
- Simpsons Rule
- Integral of a Trigonometric Function 1
- Integral of a Trigonometric Function 2
- Applications of Integration for Trig Functions