# Solve Integral & Complex Exponential Problems: Help Needed

• Geronimo85
For 2) use De Moivre's theorem to find the three solutions.In summary, the conversation discussed two homework problems: the first being an indefinite integral involving exponential and cosine functions, and the second involving finding all values of i^(2/3). The possible solution methods were briefly mentioned, including using exponential notation for the first problem and De Moivre's theorem for the second. The three solutions for the second problem were also given.
Geronimo85
I have two homework problems that have been driving me nuts:

1.) evaluate the indefinite integral:

integral(dx(e^ax)cos^2(2bx))

where a and b are real positive constants. I just don't know where to start on it.

2.) Find all values of i^(2/3)

So far I have:

i^(2/3)
= e^(2/3*ln(i))
= e^(2/3*i*(Pi/2 + 2*n*Pi))
= e^(i*Pi/3)*e^(i*n*4Pi/3)

I know from the back of my book my three solutions should end up being (1+i*sqrt(3))/2, (1-i*sqrt(3))/2, -1. But I can't seem to get there. I'd really appreciate any help. Sorry if my shorthand is confusing.

For 1) use the exponential notation for cos x, then integrate. You can do it by integration by parts twice, then use 'the trick' but that would be pretty messy.

For the first problem, you can use the substitution u = e^ax, du = ae^ax dx to transform the integral into:

integral(cos^2(2bx)) du

Then, you can use the identity cos^2(x) = (1 + cos(2x))/2 to rewrite the integral as:

1/2 integral(1 + cos(4bx)) du

Integrating each term separately, you get:

u/2 + 1/8 sin(4bx) + C

Substituting back u = e^ax, the final solution is:

e^ax/2 + 1/8 sin(4bx) + C

For the second problem, your approach is correct. However, you need to be careful with the exponent of e. It should be:

e^(i*2Pi/3)*e^(i*n*4Pi/3)

Then, you can use Euler's formula e^(ix) = cos(x) + i*sin(x) to rewrite the expression as:

e^(i*2Pi/3) = cos(2Pi/3) + i*sin(2Pi/3) = -1/2 + i*sqrt(3)/2

e^(i*4Pi/3) = cos(4Pi/3) + i*sin(4Pi/3) = -1/2 - i*sqrt(3)/2

So, the three solutions are:

e^(i*2Pi/3) = -1/2 + i*sqrt(3)/2

e^(i*4Pi/3) = -1/2 - i*sqrt(3)/2

e^(i*0) = 1 + 0*i = 1

which can also be written as:

(1+i*sqrt(3))/2, (1-i*sqrt(3))/2, -1

## 1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is also known as the anti-derivative, as it is the reverse operation of differentiation. Integrals are used to calculate various quantities such as distance, volume, and probability in mathematics and physics.

## 2. What are complex exponential problems?

Complex exponential problems involve expressions with both real and imaginary components raised to a power. These types of problems are commonly seen in calculus and engineering, and involve using rules of exponents and logarithms to simplify the expression and solve for the unknown variable.

## 3. How do I solve an integral & complex exponential problem?

The first step in solving an integral & complex exponential problem is to simplify the expression using the rules of exponents and logarithms. Then, use integration techniques such as substitution or integration by parts to find the anti-derivative. Finally, plug in the limits of integration and evaluate the integral.

## 4. What are some tips for solving these types of problems?

Some tips for solving integral & complex exponential problems include: always simplify the expression before attempting to integrate, be familiar with the rules of exponents and logarithms, and practice using different integration techniques. It is also helpful to check your answers using the fundamental theorem of calculus.

## 5. Can you provide an example of solving an integral & complex exponential problem?

Sure, for example, we can solve the integral of e^(2x+3)dx. First, we simplify the expression to e^(2x)e^3. Then, using the rule e^(a+b) = e^a * e^b, we get e^(2x) * e^3. Next, we use the substitution u = 2x, du = 2dx, and the anti-derivative of e^u is e^u. So, the integral becomes 1/2 * e^(2x) * e^3 + C. Finally, we plug in the limits of integration and evaluate the integral to get the final answer of (e^(2x+3))/2 + C.

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