Evaluate the indefinite integral \int \frac {du}{4\sqrt u}

Question:

Evaluate the indefinite integral {eq}\int \frac {du}{4\sqrt u} {/eq}

Indefinite Integral in Calculus:

We can rewrite the given type of function as a power function by using the rule {eq}\sqrt[n]a^m= a^{m/n}. {/eq} . Integration is used to solve many type of problems in mathematics.

The integral power rule is one of the most commonly used integration rule, which states that {eq}\int u^n dx = \dfrac{u^{n+1}}{n+1}+C {/eq}.

Answer and Explanation:

We are given:

{eq}\displaystyle \int \frac {du}{4\sqrt u} {/eq}


Simplify:

{eq}= \displaystyle \int \frac {1}{4 } u^{-1/2} \ du {/eq}


Take the constant out:

{eq}= \displaystyle \frac {1}{4} \int u^{-1/2} \ du {/eq}


Apply integral power rule:

{eq}=\displaystyle \frac {1}{4 } \dfrac{ u^{-\frac{1}{2}+1} }{-\frac{1}{2}+1}+C {/eq}

{eq}=\displaystyle \dfrac{ 2u^{\frac{1}{2}} }{4}+C \quad \text{where C is an arbitrary constant.} {/eq}

{eq}=\displaystyle \dfrac{ \sqrt u }{2}+C {/eq}


Therefore the solution is:

{eq}\displaystyle {\boxed{ \int \frac {du}{4 \sqrt u} = \dfrac{ \sqrt u }{2}+C }} {/eq}


Learn more about this topic:

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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

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