# Evaluating Indifinite Integral

• PolyFX
In summary, the integral \int \sec^{2}(3x)e^{\tan(3x)}dx can be solved using integration by substitution, with the substitution u = \tan(3x). Simplifying the resulting integral, we get the solution 1/3e^{\tan(3x)} + C, as per the professor's solution sheet.
PolyFX

## Homework Statement

$$\int \sec^{2}(3x)e^{\tan(3x)}dx$$

## Homework Equations

The method I am trying to use is integration by substitution(u substitution).

## The Attempt at a Solution

I start out by making u = tan(3x)

So i end up having $$\int \sec^{2}(3x)e^{u}dx$$

Stuck after here though. How do I simplify this further?

The final solution as per my professors solution sheet should be $$1/3e^{tan(3x)} + C$$

However, I cannot seem to figure out why this is the solution.

-Thank you

change sec^{2} 3x into 1 + tan^{2}

Then make substitutions

If u=tan(3x), then what's du?

Unto said:
change sec^{2} 3x into 1 + tan^{2}

Then make substitutions
This is not a helpful suggestion.

Would du be $$sec^{2}(3x) (3dx)$$?

Yes, it would. Can you use that to finish? Solve that for dx and put it into the original integral.

hi, sorry for the late reply.

Solving for dx I got

dx= du/(Sec^2(3x)) x 3

So

Sec^2(3x)e^u du/sec^2(3x) x 3

sec^2(3x) cancel each other out?

so I'm left with

e^u(du)(1/3) = 1/3e^u(du)

so 1/3e^(tan(3x))+C?

Is my approach correct?

-Thank You

PolyFX said:
hi, sorry for the late reply.

Solving for dx I got

dx= du/(Sec^2(3x)) x 3

So

Sec^2(3x)e^u du/sec^2(3x) x 3

sec^2(3x) cancel each other out?

so I'm left with

e^u(du)(1/3) = 1/3e^u(du)

so 1/3e^(tan(3x))+C?

Is my approach correct?

-Thank You

Yes, yes, yes.

## What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the inverse operation of differentiation. It is used to calculate the antiderivative of a function.

## What is the difference between an indefinite integral and a definite integral?

The main difference between an indefinite integral and a definite integral is that the former gives a general solution while the latter gives a specific value. An indefinite integral includes a constant term, while a definite integral has upper and lower limits of integration.

## How do you evaluate an indefinite integral?

To evaluate an indefinite integral, you need to use integration techniques such as substitution, integration by parts, or partial fractions. These techniques help to simplify the integrand and make it easier to integrate.

## What is the purpose of evaluating an indefinite integral?

Evaluating an indefinite integral is useful in finding the original function from its derivative, determining the area under a curve, and solving problems in physics, engineering, and economics.

## What are some common mistakes to avoid when evaluating an indefinite integral?

Some common mistakes to avoid when evaluating an indefinite integral include forgetting to add the constant of integration, using incorrect integration techniques, and making algebraic errors while simplifying the integrand.

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