Multiple problems to solve by means of integration

[pavadrin]

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,
Pavadrin

 

[neutrino]

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?

 

[pavadrin]

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.

 

[radou]

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 $$\int \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan (\frac{x}{a})+C$$.

 

[neutrino]

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. $$x = a\tan{\theta}$$.