Blog: `int cos(sqrt(x)) dx` First make a substitution and then use integration by parts to evaluate the integral

Monday, June 9, 2014

$\displaystyle\int{\cos{{\left(\sqrt{{{x}}}\right)}}}{\left.{d}{x}\right.}$ First make a substitution and then use integration by parts to evaluate the integral

You need to use the following substitution, such that:

$\displaystyle\sqrt{{x}}={t}\Rightarrow\frac{{{\left.{d}{x}\right.}}}{{{2}\sqrt{{x}}}}={\left.{d}{t}\right.}\Rightarrow{\left.{d}{x}\right.}={2}{t}{\left.{d}{t}\right.}$

Replacing the variable yields:

$\displaystyle\int{\cos{\sqrt{{x}}}}{\left.{d}{x}\right.}=\int{\left({\cos{{t}}}\right)}\cdot{\left({2}{t}{\left.{d}{t}\right.}\right)}$

You need to use the formula of integration by parts, such that:

$\displaystyle\int{u}{d}{v}={u}{v}-\int{v}{d}{u}$

$\displaystyle{u}={t}\Rightarrow{d}{u}={\left.{d}{t}\right.}$

$\displaystyle{d}{v}={\cos{{t}}}\Rightarrow{v}=\int{\cos{{t}}}{\left.{d}{t}\right.}={\sin{\ldots}}$

You need to use the following substitution, such that:

$\displaystyle\sqrt{{x}}={t}\Rightarrow\frac{{{\left.{d}{x}\right.}}}{{{2}\sqrt{{x}}}}={\left.{d}{t}\right.}\Rightarrow{\left.{d}{x}\right.}={2}{t}{\left.{d}{t}\right.}$

Replacing the variable yields:

$\displaystyle\int{\cos{\sqrt{{x}}}}{\left.{d}{x}\right.}=\int{\left({\cos{{t}}}\right)}\cdot{\left({2}{t}{\left.{d}{t}\right.}\right)}$

You need to use the formula of integration by parts, such that:

$\displaystyle\int{u}{d}{v}={u}{v}-\int{v}{d}{u}$

$\displaystyle{u}={t}\Rightarrow{d}{u}={\left.{d}{t}\right.}$

$\displaystyle{d}{v}={\cos{{t}}}\Rightarrow{v}=\int{\cos{{t}}}{\left.{d}{t}\right.}={\sin{{t}}}$

$\displaystyle\int{t}\cdot{\cos{{t}}}{\left.{d}{t}\right.}={t}\cdot{\sin{{t}}}-\int{\sin{{t}}}{\left.{d}{t}\right.}$

$\displaystyle\int{t}\cdot{\cos{{t}}}{\left.{d}{t}\right.}={t}\cdot{\sin{{t}}}+{\cos{{t}}}+{C}$

$\displaystyle{2}\int{t}\cdot{\cos{{t}}}{\left.{d}{t}\right.}={2}{t}\cdot{\sin{{t}}}+{2}{\cos{{t}}}+{C}$

Replacing back the variable $\displaystyle\sqrt{{x}}$ for t, yields:

$\displaystyle\int{\cos{\sqrt{{x}}}}{\left.{d}{x}\right.}={2}\sqrt{{x}}\cdot{\sin{{\left(\sqrt{{x}}\right)}}}+{2}{\cos{{\left(\sqrt{{x}}\right)}}}+{C}$

Hence, evaluating the integral, using substitution and integration by parts yields $\displaystyle\int{\cos{\sqrt{{x}}}}{\left.{d}{x}\right.}={2}\sqrt{{x}}\cdot{\sin{{\left(\sqrt{{x}}\right)}}}+{2}{\cos{{\left(\sqrt{{x}}\right)}}}+{C}.$

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Monday, June 9, 2014

$\displaystyle\int{\cos{{\left(\sqrt{{{x}}}\right)}}}{\left.{d}{x}\right.}$ First make a substitution and then use integration by parts to evaluate the integral

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What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?

Monday, June 9, 2014

First make a substitution and then use integration by parts to evaluate the integral

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What is the Exposition, Rising Action, Climax, and Falling Action of &quot;One Thousand Dollars&quot;?

$\displaystyle\int{\cos{{\left(\sqrt{{{x}}}\right)}}}{\left.{d}{x}\right.}$ First make a substitution and then use integration by parts to evaluate the integral

What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?