`int (x^4 + 5^x) dx` Find the indefinite integral

Monday, December 17, 2012

$\displaystyle\int{\left({{x}}^{{4}}+{{5}}^{{x}}\right)}{\left.{d}{x}\right.}$ Find the indefinite integral

For the given integral: $\displaystyle\int{\left({{x}}^{{4}}+{{5}}^{{x}}\right)}{\left.{d}{x}\right.}$ , we may apply the basic integration property:

$\displaystyle\int{\left({u}+{v}\right)}{\left.{d}{x}\right.}=\int{\left({u}\right)}{\left.{d}{x}\right.}+\int{\left({v}\right)}{\left.{d}{x}\right.}$ .

We can integrate each term separately.

$\displaystyle\int{\left({{x}}^{{4}}+{{5}}^{{x}}\right)}{\left.{d}{x}\right.}=\int{\left({{x}}^{{4}}\right)}{\left.{d}{x}\right.}+\int{\left({{5}}^{{x}}\right)}{\left.{d}{x}\right.}$

For the integration of first term: $\displaystyle\int{\left({{x}}^{{4}}\right)}{\left.{d}{x}\right.}$ ,

we apply the Power Rule for integration:

$\displaystyle\int{\left({{x}}^{{n}}\right)}{\left.{d}{x}\right.}=\frac{{{x}}^{{{n}+{1}}}}{{{n}+{1}}}+{C}$ .

Then,

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

$\displaystyle\int{\left({{x}}^{{4}}\right)}{\left.{d}{x}\right.}=\frac{{{x}}^{{{5}}}}{{{5}}}+{C}$

For the integration of first term: $\displaystyle\int{\left({{5}}^{{x}}\right)}{\left.{d}{x}\right.}$ , we apply the basic integration formula for exponential function :

$\displaystyle\int{\left({{a}}^{{x}}\right)}{\left.{d}{x}\right.}=\frac{{{a}}^{{x}}}{{\ln{{\left({a}\right)}}}}+{C}$ where $\displaystyle{a}\ne{1}$

Then,

$\displaystyle\int{\left({{5}}^{{x}}\right)}{\left.{d}{x}\right.}=\frac{{{5}}^{{x}}}{{\ln{{\left({5}\right)}}}}+{C}$

Combining the two integrations for the final answer:

$\displaystyle\int{\left({{x}}^{{4}}+{{5}}^{{x}}\right)}{\left.{d}{x}\right.}=\frac{{{x}}^{{{5}}}}{{{5}}}+\frac{{{x}}^{{{5}}}}{{{5}}}+{C}$

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Monday, December 17, 2012

$\displaystyle\int{\left({{x}}^{{4}}+{{5}}^{{x}}\right)}{\left.{d}{x}\right.}$ 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?

Monday, December 17, 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?

$\displaystyle\int{\left({{x}}^{{4}}+{{5}}^{{x}}\right)}{\left.{d}{x}\right.}$ 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?