`int (x + 4)6^((x+4)^2) dx` Find the indefinite integral

Friday, May 4, 2012

$\displaystyle\int{\left({x}+{4}\right)}{{6}}^{{{{\left({x}+{4}\right)}}^{{2}}}}{\left.{d}{x}\right.}$ Find the indefinite integral

$\displaystyle\int{\left({x}+{4}\right)}{{6}}^{{{{\left({x}+{4}\right)}}^{{2}}}}{\left.{d}{x}\right.}=$

We will make substitution $\displaystyle{u}={{\left({x}+{4}\right)}}^{{2}}.$ Differentiation the substitution gives us $\displaystyle{d}{u}={2}{\left({x}+{4}\right)}{\left.{d}{x}\right.}.$

Now we rewrite the integral.

$\displaystyle\frac{{1}}{{2}}\int{{6}}^{{{{\left({x}+{4}\right)}}^{{2}}}}{2}{\left({x}+{4}\right)}{\left.{d}{x}\right.}=$

The above integral is equal to the starting one because $\displaystyle\frac{{1}}{{2}}$ and $\displaystyle{2}$ cancel out. Now we use the substitution while.

$\displaystyle\frac{{1}}{{2}}\int{{6}}^{{u}}{d}{u}=\frac{{1}}{{2}}\cdot\frac{{{6}}^{{u}}}{{\ln{{6}}}}+{C}$

To write the final solution we simply return the substitution i.e. we put $\displaystyle{{\left({x}+{4}\right)}}^{{2}}$ instead of $\displaystyle{u}.$

$\displaystyle\frac{{{6}}^{{{{\left({x}+{4}\right)}}^{{2}}}}}{{{2}{\ln{{6}}}}}+{C}$ where C is a constant

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Friday, May 4, 2012

$\displaystyle\int{\left({x}+{4}\right)}{{6}}^{{{{\left({x}+{4}\right)}}^{{2}}}}{\left.{d}{x}\right.}$ Find the indefinite integral

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Friday, May 4, 2012

Find the indefinite integral

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Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle\int{\left({x}+{4}\right)}{{6}}^{{{{\left({x}+{4}\right)}}^{{2}}}}{\left.{d}{x}\right.}$ Find the indefinite integral

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?