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Integral Steps:
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There are multiple ways to do this integral.
Method #1
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Let [tex]u = \sqrt{x}[/tex].
Then let [tex]du = \frac{dx}{2 \sqrt{x}}[/tex] and substitute [tex]2 du[/tex]:
[tex]\int 2 u \log{\left (u \right )}\, du[/tex]
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The integral of a constant times a function is the constant times the integral of the function:
[tex]\int u \log{\left (u \right )}\, du = 2 \int u \log{\left (u \right )}\, du[/tex]
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Use integration by parts:
Let [tex]u{\left (u \right )} = \log{\left (u \right )}[/tex] and let [tex]\operatorname{dv}{\left (u \right )} = u[/tex].
Then [tex]\operatorname{du}{\left (u \right )} = \frac{1}{u}[/tex].
To find [tex]v{\left (u \right )}[/tex]:
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The integral of [tex]u^{n}[/tex] is [tex]\frac{u^{n + 1}}{n + 1}[/tex] when [tex]n \neq -1[/tex]:
[tex]\int u\, du = \frac{u^{2}}{2}[/tex]
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
[tex]\int \frac{u}{2}\, du = \frac{1}{2} \int u\, du[/tex]
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The integral of [tex]u^{n}[/tex] is [tex]\frac{u^{n + 1}}{n + 1}[/tex] when [tex]n \neq -1[/tex]:
[tex]\int u\, du = \frac{u^{2}}{2}[/tex]
So, the result is: [tex]\frac{u^{2}}{4}[/tex]
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So, the result is: [tex]u^{2} \log{\left (u \right )} - \frac{u^{2}}{2}[/tex]
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Now substitute [tex]u[/tex] back in:
[tex]x \log{\left (\sqrt{x} \right )} - \frac{x}{2}[/tex]
Method #2
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Use integration by parts:
Let [tex]u{\left (x \right )} = \log{\left (\sqrt{x} \right )}[/tex] and let [tex]\operatorname{dv}{\left (x \right )} = 1[/tex].
Then [tex]\operatorname{du}{\left (x \right )} = \frac{1}{2 x}[/tex].
To find [tex]v{\left (x \right )}[/tex]:
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The integral of a constant is the constant times the variable of integration:
[tex]\int 1\, dx = x[/tex]
Now evaluate the sub-integral.
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The integral of a constant is the constant times the variable of integration:
[tex]\int \frac{1}{2}\, dx = \frac{x}{2}[/tex]
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Add the constant of integration:
[tex]x \log{\left (\sqrt{x} \right )} - \frac{x}{2}+ \mathrm{constant}[/tex]
The answer is:
[tex]x \log{\left (\sqrt{x} \right )} - \frac{x}{2}+ \mathrm{constant}[/tex]
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* is multiplication
oo is $\infty$
pi is $\pi$
x^2 is x2
sqrt(x) is $\sqrt{x}$
sqrt[3](x) is $\sqrt[3]{x}$
(a+b)/(c+d) is $\frac{a+b}{c+d}$
Common Integrals
$\int 0dx = \text{const}$
$\int\ dx = x + \text{const}$
$\int kdx = kx + \text{const}$
$\int x^n\ dx = \frac{1}{n+1}x^{n+1} + \text{const}$ here n≠-1
$\int \frac{1}{x}\ dx = \int x^{-1}\ dx = \ln|x| + \text{const}$
$\int x^{-n}\ dx = \frac{1}{-n+1}x^{-n+1} + \text{const}$
$\int \frac{1}{ax+b}\ dx = \frac{1}{a}\ln|ax+b| + \text{const}$
$\int e^x\ dx = e^x + \text{const}$
$\int a^x\ dx = \frac{a^x}{\\ln a} + \text{const}$
$\int \sin(x)\ dx = -\cos(x) + \text{const}$
$\int \cos(x)\ dx = \sin(x) + \text{const}$
$\int \tan(x)\ dx = \ln|sec(x)| + \text{const}$
$\int \cot(x)\ dx = \ln|\sin(x)| + \text{const}$
$\int \frac{1}{\sqrt{1-x^2}} \ dx = \arcsin(x) + \text{const}$
$\int -\frac{1}{\sqrt{1-x^2}} \ dx = \arccos(x) + \text{const}$
$\int \frac{1}{1+ x^2}\ dx = \arctan(x) + \text{const}$
$\int -\frac{1}{1+x^2}\ dx = \text{arccot}(x) + \text{const}$
Integration by Parts
$\int u\ dv = uv - \int v\ du$
$\int\limits_{a}^{b} u\ dv = uv |_a^b - \int v\ du$
Trigonometric Substitutions
$\sqrt{a^2 - b^2x^2}$ $\Rightarrow x=\frac{a}{b}\sin\theta$ and $\cos^2\theta = 1 - \sin^2\theta$
$\sqrt{a^2 + b^2x^2}$ $\Rightarrow x=\frac{a}{b}\tan\theta$ and $\sec^2\theta = 1 + \tan^2\theta$
$\sqrt{b^2x^2 - a^2}$ $\Rightarrow x=\frac{a}{b}\sec\theta$ and $\tan^2\theta = \sec^2\theta - 1$
