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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 [tex]\int {xe^{x^2 } } \;dx[/tex]

This problem I simply do not know how to solve.

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Find the indefinite integral of [tex]\int {\frac{7}{{x^2 + 4x + 12}}} \;dx[/tex]

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.

[tex]

\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}

[/tex]

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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

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# Multiple problems to solve by means of integration

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