# Multiple problems to solve by means of integration

In summary, the conversation discusses a request for help with solving problems involving integration and the process of finding indefinite integrals of given functions. The conversation also includes a discussion on the integral of 1/(x^2+a^2) and how to derive it using a substitution method.
Hey
Recently I have been given multiple problems to solve by means of integration. There are some problems which I am unsure on how to go about solving, or just don’t know. All help is greatly appreciated.

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Find the indefinite integral of $$\int {xe^{x^2 } } \;dx$$

This problem I simply do not know how to solve.

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Find the indefinite integral of $$\int {\frac{7}{{x^2 + 4x + 12}}} \;dx$$

This problem I began by trying to find factors of denominator of the fraction however found none. My reasoning on how to go about solving this problem is shown below, however I am unsure about my final answer.

$$\begin{array}{l} f\left( x \right) = \int {\frac{7}{{x^2 + 4x + 12}}} \;dx \\ \int {\frac{7}{{\left( {x + 2} \right)^2 + 8}}} \;dx \\ \int {\frac{7}{{u^2 + 8}}} \;dx\quad \quad u = x + 2 \\ u' = \frac{{du}}{{dx}} \\ dx = \frac{{du}}{{u'}} \\ dx = \frac{{du}}{1} \\ \int {\frac{7}{{u^2 + 8}}} \;du \\ 7\int {\frac{1}{{u^2 + 8}}\,du} \\ f\left( x \right) = \ln \left| {u^2 + 8x} \right| + c \\ f\left( x \right) = \ln \left| {\left( {x + 2} \right)^2 + 8x} \right| + c \\ f\left( x \right) = \ln \left| {x^2 + 12x + 4} \right| + c \\ \end{array}$$

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There are others, but I'll only post these for now. I'll go back and try the others again and I may not need to post them here. Thanks in advance again,

For the first do the sub. u = x^2

As for the second problem, the integral of 1/(u^+8) is not ln(u^+8). One way to check if you've got the right answer, is to take the derivative of the result and see if you obtain the integrand.

int(1/(u^+8)) is quite easy. Do you know the integral of 1/(x^2+a^2), where 'a' is a constant?

I am unfamiliar with the integral of 1/(x^2+a^2), where 'a' is a constant, could you please explain how to derive this integral. Thanks.

I am unfamiliar with the integral of 1/(x^2+a^2), where 'a' is a constant, could you please explain how to derive this integral. Thanks.

It is a 'table integral', and it equals $$\int \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan (\frac{x}{a})+C$$.

I am unfamiliar with the integral of 1/(x^2+a^2), where 'a' is a constant, could you please explain how to derive this integral. Thanks.
For that use the sub. $$x = a\tan{\theta}$$.

## What is integration?

Integration is a mathematical process that involves finding the area under a curve or the accumulation of a quantity over a given interval. It is a fundamental concept in calculus and is used to solve a wide range of problems in various fields, including science, engineering, and economics.

## What are some common problems that can be solved using integration?

Integration can be used to solve a variety of problems, such as finding the volume of irregularly shaped objects, calculating the work done by a variable force, determining the center of mass of an object, and solving differential equations in physics and engineering.

## What are the different methods of integration?

There are several methods of integration, including the basic integration rules, such as the power rule, substitution rule, and integration by parts, as well as more advanced techniques like trigonometric substitution, partial fractions, and numerical integration methods.

## How can integration be helpful in solving multiple problems?

Integration is a powerful tool that allows us to break down complex problems into smaller, more manageable parts and solve them step by step. By using integration, we can solve multiple problems simultaneously and find the most efficient and accurate solutions.

## What are some tips for effectively using integration to solve problems?

Some tips for using integration to solve problems include identifying the type of problem and choosing the appropriate integration method, carefully setting up the problem and labeling all variables, and checking the solution for accuracy by differentiating it to see if it matches the original function.

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