New Notes Blog: `int e^(sqrt(2x)) dx` Find the indefinite integral by using substitution followed by integration by parts.

Monday, 25 November 2013

$\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}$ Find the indefinite integral by using substitution followed by integration by parts.

To evaluate the given integral problem $\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}$ us u-substituion, we may let:

$\displaystyle{u}={2}{x}$ then $\displaystyle{d}{u}={2}{\left.{d}{x}\right.}$ or $\displaystyle\frac{{{d}{u}}}{{2}}={\left.{d}{x}\right.}$ .

Plug-in the values $\displaystyle{u}={2}{x}$ and $\displaystyle{\left.{d}{x}\right.}=\frac{{{d}{u}}}{{2}}$ , we get:

$\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}=\int{{e}}^{{\sqrt{{{u}}}}}\cdot\frac{{{d}{u}}}{{2}}$

Apply the basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .

$\displaystyle\int{{e}}^{{\sqrt{{{u}}}}}\cdot\frac{{{d}{u}}}{{2}}=\frac{{1}}{{2}}\int{{e}}^{{\sqrt{{{u}}}}}{d}{u}$

Apply another set of substitution, we let:

$\displaystyle{w}=\ldots$

To evaluate the given integral problem $\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}$ us u-substituion, we may let:

$\displaystyle{u}={2}{x}$ then $\displaystyle{d}{u}={2}{\left.{d}{x}\right.}$ or $\displaystyle\frac{{{d}{u}}}{{2}}={\left.{d}{x}\right.}$ .

Plug-in the values $\displaystyle{u}={2}{x}$ and $\displaystyle{\left.{d}{x}\right.}=\frac{{{d}{u}}}{{2}}$ , we get:

$\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}=\int{{e}}^{{\sqrt{{{u}}}}}\cdot\frac{{{d}{u}}}{{2}}$

Apply the basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .

$\displaystyle\int{{e}}^{{\sqrt{{{u}}}}}\cdot\frac{{{d}{u}}}{{2}}=\frac{{1}}{{2}}\int{{e}}^{{\sqrt{{{u}}}}}{d}{u}$

Apply another set of substitution, we let:

$\displaystyle{w}=\sqrt{{{u}}}$

Square both sides of $\displaystyle{w}=\sqrt{{{u}}}$ , we get: $\displaystyle{{w}}^{{2}}={u}$

Take the derivative on each side, it becomes: $\displaystyle{2}{w}{d}{w}={d}{u}$

Plug-in $\displaystyle{w}=\sqrt{{{u}}}$ and $\displaystyle{d}{u}={2}{w}{d}{w}$ , we get:

$\displaystyle\frac{{1}}{{2}}\int{{e}}^{{\sqrt{{{u}}}}}{d}{u}=\frac{{1}}{{2}}\int{{e}}^{{{w}}}\cdot{2}{w}{d}{w}$

$\displaystyle=\frac{{1}}{{2}}\cdot{2}\int{{e}}^{{{w}}}\cdot{w}{d}{w}$

$\displaystyle=\int{{e}}^{{w}}\cdot{w}{d}{w}$ .

To evaluate the integral further, we apply integration by parts: $\displaystyle\int{f{\cdot}}{g{'}}={f{\cdot}}{g{-}}\int{g{\cdot}}{f{'}}$

Let: $\displaystyle{f{=}}{w}$ then $\displaystyle{f{'}}={d}{w}$

$\displaystyle{g{'}}={{e}}^{{w}}{d}{w}$ then $\displaystyle{g{=}}{{e}}^{{w}}$

Applying the formula for integration by parts, we get:

$\displaystyle\int{{e}}^{{w}}\cdot{w}{d}{w}={w}\cdot{{e}}^{{w}}-\int{{e}}^{{w}}{d}{w}$

$\displaystyle={w}{{e}}^{{w}}-{{e}}^{{w}}+{C}$

Recall we let: $\displaystyle{w}=\sqrt{{{u}}}$ and $\displaystyle{u}={2}{x}$ then $\displaystyle{w}=\sqrt{{{2}{x}}}$ .

Plug-in $\displaystyle{w}=\sqrt{{{2}{x}}}$ on $\displaystyle{w}{{e}}^{{w}}-{{e}}^{{w}}+{C}$ , we get the complete indefinite integral as:

$\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}=\sqrt{{{2}{x}}}{{e}}^{{\sqrt{{{2}{x}}}}}-{{e}}^{{\sqrt{{{2}{x}}}}}+{C}$

New Notes Blog

Monday, 25 November 2013

$\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}$ Find the indefinite integral by using substitution followed by integration by parts.

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Monday, 25 November 2013

Find the indefinite integral by using substitution followed by integration by parts.

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle\int{{e}}^{{\sqrt{{{2}{x}}}}}{\left.{d}{x}\right.}$ Find the indefinite integral by using substitution followed by integration by parts.

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?