Find the integral using the tabular method

Integration by Parts using Tabular Approach

Let’s find the integral of x²eˣ using the tabular method:

Choose Functions:

u = x²
dv = eˣ dx

Create a Table:

For the table, alternate between differentiating u and integrating dv. Start by listing u, its successive derivatives, and dv, its successive integrals:

u	dv
x²	eˣ
2x	eˣ
2	eˣ

Apply Signs and Multiply Diagonally:

Now, apply alternating signs down the table. Multiply the terms diagonally and add them up:

(+) × (x²eˣ) + (-) × (2xeˣ) + (+) × (2eˣ)

Here’s how the multiplication works:

First row: (+) × (x²eˣ) = x²eˣ
Second row: (-) × (2xeˣ) = -2xeˣ
Third row: (+) × (2eˣ) = 2eˣ

Final Result:

The integral of x²eˣ is:

∫ x²eˣ dx = x²eˣ – 2xeˣ + 2eˣ + C

where C is the constant of integration.

August 9, 2023August 9, 2023

Find the integral of xeˣ using the tabular method

Integration by Parts using Tabular Approach

Let’s find the integral of xeˣ using the tabular method:

1. Choose Functions:
  - u = x
  - dv = eˣ dx
2. Create a Table:

For the table, alternate between differentiating u and integrating dv. Start by listing u, its successive derivatives, and dv, its successive integrals:

u	dv
x	eˣ
1	eˣ
0	eˣ

1. Apply Signs and Multiply Diagonally:

Now, apply alternating signs down the table. Multiply the terms diagonally and add them up:

(+) × (xeˣ) + (-) × (eˣ) + (+) × (eˣ)

Here’s how the multiplication works:

1. - First row: (+) × (xeˣ) = xeˣ
  - Second row: (-) × (eˣ) = -eˣ
  - Third row: (+) × (eˣ) = eˣ
2. Final Result:

The integral of xeˣ is:

∫ xeˣ dx = xeˣ – eˣ + C

where C is the constant of integration.

August 9, 2023August 9, 2023

Integration by Parts using Tabular Approach

Given the integral:

$$ \int x^2 \ln(x) \,dx $$

We’ll use the tabular method to solve it:

Choose Functions:

$$ u = \ln(x) $$
$$ dv = x^2 \,dx $$

Create a Table:

For the table, we will alternate between differentiating $u$ and integrating $dv$. Start by listing $u$, its successive derivatives, and $dv$, its successive integrals:

$$ u $$	$$ dv $$
$$ \ln(x) $$	$$ x^2 $$
$$ \frac{1}{x} $$	$$ \frac{x^3}{3} $$
$$ -\frac{1}{x^2} $$	$$ \frac{x^4}{12} $$
$$ \frac{2}{x^3} $$	$$ \frac{x^5}{60} $$

Apply Signs and Multiply Diagonally:

Now, apply alternating signs down the table. Multiply the terms diagonally and add them up:

$$ (+) \cdot (\ln(x) \cdot x^2) + (-) \cdot \left(\frac{1}{x} \cdot \frac{x^3}{3}\right) + (+) \cdot \left(-\frac{1}{x^2} \cdot \frac{x^4}{12}\right) + (-) \cdot \left(\frac{2}{x^3} \cdot \frac{x^5}{60}\right) $$

Here’s how the multiplication works:

First row: $$ (+) \cdot (\ln(x) \cdot x^2) = x^2 \ln(x) $$
Second row: $$ (-) \cdot \left(\frac{1}{x} \cdot \frac{x^3}{3}\right) = -\frac{x^2}{3} $$
Third row: $$ (+) \cdot \left(-\frac{1}{x^2} \cdot \frac{x^4}{12}\right) = \frac{x^2}{12} $$
Fourth row: $$ (-) \cdot \left(\frac{2}{x^3} \cdot \frac{x^5}{60}\right) = -\frac{x^2}{30} $$

Simplify and Add Up Terms:

$$ \int x^2 \ln(x) \,dx = x^2 \ln(x) – \frac{x^2}{3} – \frac{x^2}{12} – \frac{x^2}{30} + C $$

Final Result:

$$ \int x^2 \ln(x) \,dx = x^2 \ln(x) – \frac{13x^2}{60} + C $$

August 9, 2023August 9, 2023

Integration by Parts using Tabular Approach xsin(x)

Integration by Parts using Tabular Approach

Given the integral:

∫ xsin(x) dx

We’ll use the tabular method to solve it:

Choose Functions:

u = x
dv = sin(x) dx

Create a Table:

For the table, alternate between differentiating u and integrating dv. Start by listing u, its successive derivatives, and dv, its successive integrals:

u	dv
x	sin(x)
1	-cos(x)
0	-sin(x)
0	cos(x)

Apply Signs and Multiply Diagonally:

Now, apply alternating signs down the table. Multiply the terms diagonally and add them up:

(+) × (xsin(x)) + (-) × (1 × -cos(x)) + (+) × (0 × -sin(x)) + (-) × (0 × cos(x))

Here’s how the multiplication works:

First row: (+) × (xsin(x)) = xsin(x)
Second row: (-) × (1 × -cos(x)) = cos(x)
Third row: (+) × (0 × -sin(x)) = 0
Fourth row: (-) × (0 × cos(x)) = 0

Final Result:

∫ xsin(x) dx = xsin(x) + cos(x) + C

where C is the constant of integration.

August 8, 2023August 8, 2023

Trigonometric Values of Key Angles in the Unit Circle: Explained and Illustrated

Welcome to our comprehensive guide on understanding trigonometric values in the unit circle. Delve into the world of angles, coordinates, and functions as we explore the fundamental concepts that underlie trigonometry. Discover how to read the unit circle, compute cosine, sine, tangent, secant, cotangent, and cosecant values, and gain insights into their significance in mathematics, physics, and engineering.

Understanding Trigonometric Values: Angle $ \frac{\pi}{6} $ (30 Degrees)

When examining the angle $ \frac{\pi}{6} $ (30 degrees) in the first quadrant of the unit circle, you can extract its trigonometric values as follows:

Cosine ($ \cos $): The cosine value is the x-coordinate of the point on the unit circle. For $ \frac{\pi}{6} $, the x-coordinate is $ \frac{\sqrt{3}}{2} $. Therefore, $ \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} $.
Sine ($ \sin $): The sine value is the y-coordinate of the point on the unit circle. For $ \frac{\pi}{6} $, the y-coordinate is $ \frac{1}{2} $. Hence, $ \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} $.
Tangent ($ \tan $): The tangent value is the ratio of sine to cosine: $ \tan \left( \frac{\pi}{6} \right) = \frac{\sin \left( \frac{\pi}{6} \right)}{\cos \left( \frac{\pi}{6} \right)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{\sqrt{3}}{3} $.
Secant ($ \sec $): The secant value is the reciprocal of cosine: $ \sec \left( \frac{\pi}{6} \right) = \frac{1}{\cos \left( \frac{\pi}{6} \right)} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} $.
Cotangent ($ \cot $): The cotangent value is the reciprocal of tangent: $ \cot \left( \frac{\pi}{6} \right) = \frac{1}{\tan \left( \frac{\pi}{6} \right)} = \frac{3}{\sqrt{3}} = \sqrt{3} $.
Cosecant ($ \csc $): The cosecant value is the reciprocal of sine: $ \csc \left( \frac{\pi}{6} \right) = \frac{1}{\sin \left( \frac{\pi}{6} \right)} = \frac{2}{1} = 2 $.

By understanding these calculations, you can gain insight into the trigonometric relationships associated with the angle $ \frac{\pi}{6} $ and apply them to various problems involving angles in the first quadrant.

For a broader understanding of the unit circle and its implications, refer to our detailed explanation on Understanding the Unit Circle.

Understanding Trigonometric Values: Angle $ \frac{3\pi}{4} $ (135 Degrees)

When examining the angle $ \frac{3\pi}{4} $ (135 degrees) in the second quadrant of the unit circle, the trigonometric values can be derived as follows:

Cosine ($ \cos $): The cosine value is the x-coordinate of the point on the unit circle. For $ \frac{3\pi}{4} $, the x-coordinate is $ -\frac{\sqrt{2}}{2} $. Therefore, $ \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} $.
Sine ($ \sin $): The sine value is the y-coordinate of the point on the unit circle. For $ \frac{3\pi}{4} $, the y-coordinate is $ \frac{\sqrt{2}}{2} $. Hence, $ \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} $.
Tangent ($ \tan $): The tangent value is the ratio of sine to cosine: $ \tan \left( \frac{3\pi}{4} \right) = \frac{\sin \left( \frac{3\pi}{4} \right)}{\cos \left( \frac{3\pi}{4} \right)} = -1 $.
Secant ($ \sec $): The secant value is the reciprocal of cosine: $ \sec \left( \frac{3\pi}{4} \right) = \frac{1}{\cos \left( \frac{3\pi}{4} \right)} = -\sqrt{2} $.
Cotangent ($ \cot $): The cotangent value is the reciprocal of tangent: $ \cot \left( \frac{3\pi}{4} \right) = \frac{1}{\tan \left( \frac{3\pi}{4} \right)} = -1 $.
Cosecant ($ \csc $): The cosecant value is the reciprocal of sine: $ \csc \left( \frac{3\pi}{4} \right) = \frac{1}{\sin \left( \frac{3\pi}{4} \right)} = \sqrt{2} $.

Understanding these calculations enables you to analyze the trigonometric relationships associated with the angle $ \frac{3\pi}{4} $ and apply them to various scenarios involving angles in the second quadrant.

To expand your comprehension of the unit circle, refer to our detailed explanation on Understanding the Unit Circle.

Understanding Trigonometric Values: Angle $ \frac{3\pi}{2} $ (270 Degrees)

When considering the angle $ \frac{3\pi}{2} $ (270 degrees) in the third quadrant of the unit circle, the trigonometric values are as follows:

Cosine ($ \cos $): The cosine value is the x-coordinate of the point on the unit circle. For $ \frac{3\pi}{2} $, the x-coordinate is $ 0 $. Therefore, $ \cos \left( \frac{3\pi}{2} \right) = 0 $.
Sine ($ \sin $): The sine value is the y-coordinate of the point on the unit circle. For $ \frac{3\pi}{2} $, the y-coordinate is $ -1 $. Thus, $ \sin \left( \frac{3\pi}{2} \right) = -1 $.
Tangent ($ \tan $): The tangent value is undefined in this case, as the cosine is zero.
Secant ($ \sec $): The secant value is undefined since the cosine is zero.
Cotangent ($ \cot $): The cotangent value is also undefined as the sine is zero.
Cosecant ($ \csc $): The cosecant value is the reciprocal of sine: $ \csc \left( \frac{3\pi}{2} \right) = \frac{1}{\sin \left( \frac{3\pi}{2} \right)} = -1 $.

Understanding these trigonometric values aids in interpreting angles in the third quadrant of the unit circle and applying them in various mathematical and scientific contexts.

For further insights into the unit circle, refer to our comprehensive guide on Understanding the Unit Circle.

Understanding Trigonometric Values: Angle $ \frac{11\pi}{6} $ (330 Degrees)

When considering the angle $ \frac{11\pi}{6} $ (330 degrees) in the fourth quadrant of the unit circle, the trigonometric values are as follows:

Cosine ($ \cos $): The cosine value is the x-coordinate of the point on the unit circle. For $ \frac{11\pi}{6} $, the x-coordinate is $ \frac{\sqrt{3}}{2} $. Therefore, $ \cos \left( \frac{11\pi}{6} \right) = \frac{\sqrt{3}}{2} $.
Sine ($ \sin $): The sine value is the y-coordinate of the point on the unit circle. For $ \frac{11\pi}{6} $, the y-coordinate is $ -\frac{1}{2} $. Thus, $ \sin \left( \frac{11\pi}{6} \right) = -\frac{1}{2} $.
Tangent ($ \tan $): The tangent value is the ratio of sine to cosine: $ \tan \left( \frac{11\pi}{6} \right) = \frac{\sin \left( \frac{11\pi}{6} \right)}{\cos \left( \frac{11\pi}{6} \right)} = -\frac{1}{\sqrt{3}} $.
Secant ($ \sec $): The secant value is the reciprocal of cosine: $ \sec \left( \frac{11\pi}{6} \right) = \frac{1}{\cos \left( \frac{11\pi}{6} \right)} = \frac{2}{\sqrt{3}} $.
Cotangent ($ \cot $): The cotangent value is the reciprocal of tangent: $ \cot \left( \frac{11\pi}{6} \right) = \frac{1}{\tan \left( \frac{11\pi}{6} \right)} = -\sqrt{3} $.
Cosecant ($ \csc $): The cosecant value is the reciprocal of sine: $ \csc \left( \frac{11\pi}{6} \right) = \frac{1}{\sin \left( \frac{11\pi}{6} \right)} = -2 $.

Understanding these trigonometric values aids in interpreting angles in the fourth quadrant of the unit circle and applying them in various mathematical and scientific contexts.

For further insights into the unit circle, refer to our comprehensive guide on Understanding the Unit Circle.

August 8, 2023August 8, 2023

Exploring Negative Angles on the Unit Circle: Trigonometric Functions Demystified

Delve into the world of negative angles on the unit circle and unravel the secrets of trigonometric functions. This interactive guide provides insights into how negative angles are mapped onto the unit circle, showcasing their cosine and sine values. Understanding these relationships is crucial for a solid foundation in trigonometry.

Understanding the Unit Circle

The unit circle is a fundamental tool in trigonometry that relates angles to the coordinates of points on the circle. It provides a visual representation of trigonometric functions and their values for different angles. The circle has a radius of 1, making it easy to work with.

Reading the Unit Circle

In the context of trigonometry, each angle corresponds to a point on the unit circle. The x-coordinate of the point represents the value of the cosine function at that angle, while the y-coordinate represents the value of the sine function.

Example 1: $-\frac{\pi}{6}$ Radians ($-30^\circ$)

For an angle of $-\frac{\pi}{6}$ radians ($-30^\circ$), the point on the unit circle is $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$. This means:

Cosine ($\cos$) of $-\frac{\pi}{6}$: $\frac{\sqrt{3}}{2}$
Sine ($\sin$) of $-\frac{\pi}{6}$: $-\frac{1}{2}$

Example 2: $-\frac{\pi}{3}$ Radians ($-60^\circ$)

For an angle of $-\frac{\pi}{3}$ radians ($-60^\circ$), the point on the unit circle is $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$. This implies:

Cosine ($\cos$) of $-\frac{\pi}{3}$: $-\frac{1}{2}$
Sine ($\sin$) of $-\frac{\pi}{3}$: $-\frac{\sqrt{3}}{2}$

Example 3: $-\frac{\pi}{2}$ Radians ($-90^\circ$)

For an angle of $-\frac{\pi}{2}$ radians ($-90^\circ$), the point on the unit circle is $(0, -1)$. This indicates:

Cosine ($\cos$) of $-\frac{\pi}{2}$: $0$
Sine ($\sin$) of $-\frac{\pi}{2}$: $-1$

Example 4: $-\frac{2\pi}{3}$ Radians ($-120^\circ$)

For an angle of $-\frac{2\pi}{3}$ radians ($-120^\circ$), the point on the unit circle is $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$. This signifies:

Cosine ($\cos$) of $-\frac{2\pi}{3}$: $-\frac{1}{2}$
Sine ($\sin$) of $-\frac{2\pi}{3}$: $-\frac{\sqrt{3}}{2}$

Understanding the unit circle empowers us to compute trigonometric function values for various angles, aiding us in solving problems across mathematics, science, and engineering.

Example 5: $-\frac{5\pi}{6}$ Radians ($-150^\circ$)

For an angle of $-\frac{5\pi}{6}$ radians ($-150^\circ$), the point on the unit circle is $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$. This means:

Cosine ($\cos$) of $-\frac{5\pi}{6}$: $-\frac{\sqrt{3}}{2}$
Sine ($\sin$) of $-\frac{5\pi}{6}$: $-\frac{1}{2}$

Example 6: $-\pi$ Radians ($-180^\circ$)

For an angle of $-\pi$ radians ($-180^\circ$), the point on the unit circle is $(-1, 0)$. This implies:

Cosine ($\cos$) of $-\pi$: $-1$
Sine ($\sin$) of $-\pi$: $0$

Example 7: $-\frac{4\pi}{3}$ Radians ($-240^\circ$)

For an angle of $-\frac{4\pi}{3}$ radians ($-240^\circ$), the point on the unit circle is $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. This signifies:

Cosine ($\cos$) of $-\frac{4\pi}{3}$: $-\frac{1}{2}$
Sine ($\sin$) of $-\frac{4\pi}{3}$: $\frac{\sqrt{3}}{2}$

Example 8: $-\frac{7\pi}{6}$ Radians ($-210^\circ$)

For an angle of $-\frac{7\pi}{6}$ radians ($-210^\circ$), the point on the unit circle is $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$. This indicates:

Cosine ($\cos$) of $-\frac{7\pi}{6}$: $-\frac{\sqrt{3}}{2}$
Sine ($\sin$) of $-\frac{7\pi}{6}$: $\frac{1}{2}$

Example 9: $-\frac{11\pi}{6}$ Radians ($-330^\circ$)

For an angle of $-\frac{11\pi}{6}$ radians ($-330^\circ$), the point on the unit circle is $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$. This means:

Cosine ($\cos$) of $-\frac{11\pi}{6}$: $\frac{\sqrt{3}}{2}$
Sine ($\sin$) of $-\frac{11\pi}{6}$: $-\frac{1}{2}$

Example 10: $-\frac{2\pi}{3}$ Radians ($-120^\circ$)

For an angle of $-\frac{2\pi}{3}$ radians ($-120^\circ$), the point on the unit circle is $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$. This implies:

Cosine ($\cos$) of $-\frac{2\pi}{3}$: $-\frac{1}{2}$
Sine ($\sin$) of $-\frac{2\pi}{3}$: $-\frac{\sqrt{3}}{2}$

Example 11: $-\frac{5\pi}{4}$ Radians ($-225^\circ$)

For an angle of $-\frac{5\pi}{4}$ radians ($-225^\circ$), the point on the unit circle is $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$. This signifies:

Cosine ($\cos$) of $-\frac{5\pi}{4}$: $-\frac{\sqrt{2}}{2}$
Sine ($\sin$) of $-\frac{5\pi}{4}$: $-\frac{\sqrt{2}}{2}$

August 8, 2023August 8, 2023

Step by Step Exercises on Finding the Modulus of a Complex Number

Welcome to our comprehensive guide on Modulus Calculation for Complex Numbers. This resource provides step-by-step instructions on how to calculate the modulus of various complex numbers, including both real and imaginary components. We cover examples such as 3 + 2i, -4 + 5i, -2 – 3i, and even pure real and imaginary numbers like 2 and 2i. Each example is broken down into easy-to-follow steps, from squaring the real and imaginary parts to taking the square root of their sum. We also provide reminders and tips, such as not including ‘i’ when calculating moduli. This guide is perfect for students, teachers, and anyone interested in enhancing their understanding of complex numbers and their moduli.

Modulus Calculation for Complex Numbers