Integral of Trigonometric Functions 2

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Question 1 of 4

Find the integral

$int tan^2 x dx$

1. $\text(tan) x+x+c$
2. $\text(tan) x-2x+c$
3. $\text(tan)^2 x-x+c$
4. $\text(tan) x-x+c$

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Integrals of Trigonometric Functions

$int \text(cos)=\text(sin)$

$int \text(sin)=-\text(cos)$

$int \text(sec)^2=\text(tan)$

Integrating Trigonometric Functions

$\int f(\color{#004ec4}{g(x)}) dx=f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$

First, convert the function into a derivable function

Take note that

$\text(sec)^2=1+\text(tan)^2x$

$\text(sec)^2$	$=$	$1+\text(tan)^2x$
$\text(sec)^2$ $-1$	$=$	$1+\text(tan)^2x$ $-1$	Subtract $1$ from both sides
$\text(sec)^2-1$	$=$	$\text(tan)^2x$

Therefore, we can use

$\text(sec)^2-1$ as a derivable substitute

Finally, substitute the components into the formula

$\int f(\color{#004ec4}{g(x)}) dx$	$=$	$f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$

$\int \text{sec}^2\color{#004ec4}{x-1}\;dx$	$=$	$\text{tan}\;x-x +c$	Substitute known values and integrate

$\text(tan) x-x+c$

Question 2 of 4

2. Question
Find the integral

$int tan x dx$
- 1. $ln (\text(sin) x)+c$
- 2. $-ln (\text(sin) x)+c$
- 3. $ln (\text(cos) x)+c$
- 4. $-ln (\text(cos) x)+c$
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Integrals of Trigonometric Functions

$int \text(cos)=\text(sin)$

$int \text(sin)=-\text(cos)$

$int \text(sec)^2=\text(tan)$

Integrating Trigonometric Functions

$\int f(\color{#004ec4}{g(x)}) dx=f(g(x))\cdot\frac{1}{\color{#004ec4}{g'(x)}} +c$

First, convert the function into a derivable function

Take note that $\text(tan) x=(\text(sin) x)/(\text(cos) x)$

Therefore, we can use $(\text(sin) x)/(\text(cos) x)$ as a derivable substitute

Next, take note that $d/(dx)\text(cos) x=-\text(sin) x$

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

$=$ $-int (-\text(sin) x)/(\text(cos) x)$

Finally, integrate the function into a natural logarithm

$-int (-\text(sin) x)/(\text(cos) x)$ $=$ $-ln (\text(cos) x)+c$

$-ln (\text(cos) x)+c$

Question 3 of 4

Find the integral

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\;\text{cos}x\;dx$

1. $sqrt2$
2. $1/(sqrt2)$
3. $1-sqrt2$
4. $1-1/(sqrt2)$

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Integrals of Trigonometric Functions

$int \text(cos)=\text(sin)$

$int \text(sin)=-\text(cos)$

$int \text(sec)^2=\text(tan)$

First, integrate the trigonometric function

$\int_{\color{#00880A}{\frac{\pi}{4}}}^{\color{#9a00c7}{\frac{\pi}{2}}}\;\text{cos}\;x\;dx$

$=$

$\bigg[\text{sin}\;x\bigg]_{\color{#00880A}{\frac{\pi}{4}}}^{\color{#9a00c7}{\frac{\pi}{2}}}$

Integrate

$\text(cos)x$

Finally, get the difference of the upper and lower limits substituted to the integral as

$x$ .

	$\bigg[\text{sin}\;x\bigg]_{\color{#00880A}{\frac{\pi}{4}}}^{\color{#9a00c7}{\frac{\pi}{2}}}$

$=$	$\sin{\color{#9a00c7}{\frac{\pi}{2}}}-\sin{\color{#00880A}{\frac{\pi}{4}}}$	Substitute the limits

$=$	$1-1/(sqrt2)$	Evaluate

$1-1/(sqrt2)$

Question 4 of 4

Find the integral

$\int_{0}^{\frac{\pi}{3}}\;3\;\text{sin}\frac{x}{2}\;dx$

1. $3sqrt3-6$
2. $-3sqrt3-6$
3. $3sqrt3+6$
4. $-3sqrt3+6$

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Integrals of Trigonometric Functions

$int \text(cos)=\text(sin)$

$int \text(sin)=-\text(cos)$

$int \text(sec)^2=\text(tan)$

First, integrate the trigonometric function

$\int_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}\;\text{sin}\;\frac{x}{2}\;dx$	$=$	$\bigg[3\left(-\text{cos}\;\frac{x}{2}\right)\bigg]_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}$	Integrate $3\text(sin) x/2$

	$=$	$\bigg[-6\text{cos}\;\frac{x}{2}\bigg]_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}$	Simplify

Finally, get the difference of the upper and lower limits substituted to the integral as

$x$ .

	$\bigg[-6\text{cos}\;\frac{x}{2}\bigg]_{\color{#00880A}{0}}^{\color{#9a00c7}{\frac{\pi}{3}}}$

$=$	$-6\;\cos{\frac{\color{#9a00c7}{\frac{\pi}{3}}}{2}}-[-6\;\cos{\frac{\color{#00880A}{0}}{2}}]$	Substitute the limits

$=$	$-6 \text(cos)(pi/6)+(6*1)$	Evaluate

$=$	$-6*(sqrt3)/2+6$	$\text(cos) pi/6=(sqrt3)/2$

$=$	$-3sqrt3+6$	Simplify

$-3sqrt3+6$

Integral of Trigonometric Functions 2

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Quiz summary

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1. Question

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Excellent!

Integrals of Trigonometric Functions

Integrating Trigonometric Functions

2. Question

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Keep Going!

Integrals of Trigonometric Functions

Integrating Trigonometric Functions

3. Question

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Fantastic!

Integrals of Trigonometric Functions

4. Question

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Well Done!

Integrals of Trigonometric Functions

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