Indefinite Integral - Use Of Trigonometric Identities

Indefinite Integral - Use of Trigonometric Identities

In calculus, an indefinite integral, also known as an antiderivative, is a function F(x) whose derivative is equal to f(x).
The symbol for the indefinite integral is ∫f(x)dx, where f(x) is the integrand and dx represents the variable of integration.
When dealing with trigonometric functions in indefinite integration, we often need to utilize trigonometric identities to simplify the integrand.
In this lecture, we will focus on the use of trigonometric identities in finding antiderivatives.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables.
These identities allow us to simplify trigonometric expressions and make it easier to perform calculations.
Some of the commonly used trigonometric identities include:
1. Pythagorean Identities: sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ
2. Reciprocal Identities: cscθ = 1/sinθ, secθ = 1/cosθ, and cotθ = 1/tanθ
3. Quotient Identities: tanθ = sinθ/cosθ and cotθ = cosθ/sinθ
4. Co-Function Identities: sin(π/2 - θ) = cosθ, cos(π/2 - θ) = sinθ, and tan(π/2 - θ) = cotθ

Example 1

Find the antiderivative of the function f(x) = 2cosx. Solution: To find the antiderivative, we can use the identity ∫cosxdx = sinx + C. Therefore, ∫2cosxdx = 2sinx + C, where C is the constant of integration.

Example 2

Find the antiderivative of the function f(x) = 3tanx. Solution: To find the antiderivative, we can use the identity ∫tanxdx = -ln|cosx| + C. Therefore, ∫3tanxdx = -3ln|cosx| + C, where C is the constant of integration.

Example 3

Find the antiderivative of the function f(x) = 4sin(2x). Solution: To find the antiderivative, we can use the identity ∫sin(ax)dx = -1/a * cos(ax) + C. Therefore, ∫4sin(2x)dx = -2cos(2x) + C, where C is the constant of integration.

Example 4

Find the antiderivative of the function f(x) = 5csc²x. Solution: To find the antiderivative, we can use the identity ∫csc²xdx = -cotx + C. Therefore, ∫5csc²xdx = -5cotx + C, where C is the constant of integration.

Example 5

Find the antiderivative of the function f(x) = 6secxtanx. Solution: To find the antiderivative, we can use the identity ∫secx tanxdx = secx + C. Therefore, ∫6secxtanxdx = 6secx + C, where C is the constant of integration.

This concludes the first 10 slides of the lecture on “Indefinite Integral - Use of Trigonometric Identities”.

Indefinite Integral - Use of Trigonometric Identities

Slides 11-20

Slide 11

Trigonometric identities play a crucial role in finding antiderivatives of trigonometric functions.
Let’s explore some more examples on how to use these identities in indefinite integration.

Slide 12

Example 6: Find the antiderivative of the function f(x) = 2sin(x)cos(x).
- Solution: We can use the identity sin(2x) = 2sin(x)cos(x) to rewrite the function as f(x) = sin(2x). The antiderivative of sin(2x) is -1/2cos(2x). Hence, the antiderivative of f(x) is -1/2cos(2x) + C.

Slide 13

Example 7: Find the antiderivative of the function f(x) = sec²(3x)tan(3x).
- Solution: To solve this, we use the identity tan(3x) = sin(3x)/cos(3x). Therefore, f(x) = sec²(3x) * (sin(3x)/cos(3x)). Simplifying, f(x) = (sin(3x)/cos²(3x)). The antiderivative of (sin(3x)/cos²(3x)) is -1/3cot(3x) + C.

Slide 14

Example 8: Find the antiderivative of the function f(x) = 4cos(5x)sin(5x).
- Solution: We can use the identity sin(2θ) = 2sin(θ)cos(θ) to rewrite the function as f(x) = 2sin(10x)cos(10x). Hence, the antiderivative of f(x) is -sin²(10x)/10 + C.

Slide 15

Example 9: Find the antiderivative of the function f(x) = (tanx + secx)².
- Solution: Expanding the square, we get f(x) = tan²x + sec²x + 2tanxsecx. The antiderivative of tan²x is -ln|cosx| + C. The antiderivative of sec²x is tanx + C. The antiderivative of 2tanxsecx is ln|secx| + C. Hence, the antiderivative of f(x) is -ln|cosx| + tanx + ln|secx| + C.

Slide 16

Example 10: Find the antiderivative of the function f(x) = (2sinx + 3cosx)/(cosx - sinx).
- Solution: To simplify this expression, we can use the quotient identity for tanx. The function can be rewritten as f(x) = (2tanx - 3)/(1 - tanx). The antiderivative of (2tanx - 3)/(1 - tanx) can be found using partial fraction decomposition, resulting in the antiderivative f(x) = -2ln|cosx| - 3x + C.

Slide 17

Recap:
- Trigonometric identities are essential tools in finding antiderivatives for trigonometric functions.
- Pythagorean identities, reciprocal identities, quotient identities, and co-function identities are commonly used.
- Examples demonstrated the use of these identities in finding antiderivatives.

</p>
</section>
<section style="font-size: 50px";>
<h2 id="indefinite-integral---use-of-trigonometric-identities-1">Indefinite Integral - Use of Trigonometric Identities</h2>
</section>
<section style="font-size: 50px";>
<h2 id="slide-21">Slide 21</h2>
<ul>
<li>Let’s continue exploring examples on the use of trigonometric identities in finding antiderivatives.</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-22">Slide 22</h2>
<ul>
<li>Example 11: Find the antiderivative of the function f(x) = sin(x)(1 - cos(x)).
<ul>
<li>Solution: Expanding the expression, we get f(x) = sin(x) - sin(x)cos(x).
The antiderivative of sin(x) is -cos(x) + C.
The antiderivative of sin(x)cos(x) can be found using integration by parts.
Applying integration by parts, we get ∫sin(x)cos(x)dx = 1/2sin²(x) + C.
Hence, the antiderivative of f(x) is -cos(x) + 1/2sin²(x) + C.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-23">Slide 23</h2>
<ul>
<li>Example 12: Find the antiderivative of the function f(x) = (2cos²(x) - 5sin(x)cos(x))/sin²(x).
<ul>
<li>Solution: We can rewrite the function using the quotient identity for cot(x).
f(x) = (2cos²(x) - 5sin(x)cos(x))/sin²(x)
= 2(1 - sin²(x))/(sin²(x)) - 5cos(x)/sin(x).
Expanding and simplifying, f(x) = 2cot²(x) - 2 - 5cot(x).
The antiderivative of cot²(x) is -x - cot(x).
The antiderivative of 5cot(x) is 5ln|sin(x)|.
Hence, the antiderivative of f(x) is -x - cot(x) - 2ln|sin(x)| + C.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-24">Slide 24</h2>
<ul>
<li>Example 13: Find the antiderivative of the function f(x) = (3tan²(x) + 4sin(x)tan²(x))/(sec²(x) - 1).
<ul>
<li>Solution: We can rewrite the function using the co-function identity for tan(π/2 - x).
f(x) = (3tan²(x) + 4sin(x)tan²(x))/(sec²(x) - 1)
= (3tan²(x) + 4sin(x)tan²(x))/(tan²(x))
= 3 + 4sin(x).
The antiderivative of 3 is 3x.
The antiderivative of 4sin(x) is -4cos(x).
Hence, the antiderivative of f(x) is 3x - 4cos(x) + C.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-25">Slide 25</h2>
<ul>
<li>Example 14: Find the antiderivative of the function f(x) = (2sin(x) + cos(x))/(sin(x) + cos(x)).
<ul>
<li>Solution: We can simplify the function using the quotient identity for tan(x).
f(x) = (2sin(x) + cos(x))/(sin(x) + cos(x))
= (2sin(x)/cos(x) + 1)/(sin(x)/cos(x) + 1)
= 2tan(x) + 1.
The antiderivative of 2tan(x) is 2ln|sec(x)|.
The antiderivative of 1 is x.
Hence, the antiderivative of f(x) is 2ln|sec(x)| + x + C.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-26">Slide 26</h2>
<ul>
<li>Example 15: Find the antiderivative of the function f(x) = (cot(x) + csc(x))/sin(x).
<ul>
<li>Solution: We can rewrite the function using the reciprocal identity for sin(x) and cot(x).
f(x) = (cot(x) + csc(x))/sin(x)
= (cos(x)/sin(x) + 1/sin(x))/(sin(x)/sin(x))
= cos(x) + csc(x).
The antiderivative of cos(x) is sin(x).
The antiderivative of csc(x) can be found using integration by parts.
Applying integration by parts, we get ∫csc(x)dx = ln|csc(x) - cot(x)|.
Hence, the antiderivative of f(x) is sin(x) + ln|csc(x) - cot(x)| + C.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-27">Slide 27</h2>
<ul>
<li>Recap:
<ul>
<li>Trigonometric identities are useful for simplifying integrands and finding antiderivatives.</li>
<li>Examples demonstrated the use of identities such as cot(x), cos(π/2 - x), and tan(π/2 - x).</li>
<li>Integration by parts can be used when identities alone are not sufficient.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-28">Slide 28</h2>
<ul>
<li>Trigonometric identities also have applications in evaluating definite integrals and solving differential equations.</li>
<li>It is important to practice applying these identities to various problems to strengthen your understanding and skills.</li>
<li>As you progress in your math studies, you will encounter more complex integrals and differential equations that require a deep understanding of trigonometric identities.</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-29">Slide 29</h2>
<ul>
<li>Let’s summarize the key takeaways from this lecture:
<ul>
<li>Trigonometric identities are powerful tools in indefinite integration.</li>
<li>They allow us to simplify integrands and find antiderivatives.</li>
<li>Identities such as cot(x), cos(π/2 - x), and tan(π/2 - x) are frequently used.</li>
<li>Integration by parts can be employed when identities alone are not sufficient.</li>
</ul>
</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="slide-30">Slide 30</h2>
<ul>
<li>Thank you for attending this lecture on “Indefinite Integral - Use of Trigonometric Identities”.</li>
<li>Practice employing these identities in various integration problems to sharpen your skills.</li>
<li>If you have any questions, feel free to ask.</li>
</ul>
</section>

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