Integrate Using u-Substitution integral of (sin( square root of x))/( square root of x) with respect to x

Solution:

Step:1

Implement exponent rules.

Step:1.1

Express $\sqrt{x}$ as $x^{\frac{1}{2}}$ using the rule $\sqrt[n]{a^x}=a^{\frac{x}{n}}$.
$\int \frac{\sin (x^{\frac{1}{2}})}{\sqrt{x}}dx$

Step:1.2

Again, express $\sqrt{x}$ as $x^{\frac{1}{2}}$.
$\int \frac{\sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}}dx$

Step:1.3

Transform $x^{\frac{1}{2}}$ in the denominator to the numerator by using the negative exponent.
$\int \sin (x^{\frac{1}{2}})(x^{\frac{1}{2}}{)}^{-1}dx$

Step:1.4

Apply exponent multiplication in $(x^{\frac{1}{2}}{)}^{-1}$.

Step:1.4.1

Utilize the exponent multiplication rule, $(a^m{)}^n=a^{mn}$.
$\int \sin (x^{\frac{1}{2}})x^{\frac{1}{2}\cdot -1}dx$

Step:1.4.2

Combine the exponents $\frac{1}{2}$ and $-1$.
$\int \sin (x^{\frac{1}{2}})x^{-\frac{1}{2}}dx$

Step:1.4.3

Place the negative sign in front of the fraction.
$\int \sin (x^{\frac{1}{2}})x^{-\frac{1}{2}}dx$

Step:2

Set $u=x^{\frac{1}{2}}$. Then $du=\frac{1}{2x^{\frac{1}{2}}}dx$, and $dx=2x^{\frac{1}{2}}du$. Substitute $u$ and $du$ into the integral.

Step:2.1

Define $u=x^{\frac{1}{2}}$ and calculate $\frac{du}{dx}$.

Step:2.1.1

Differentiate $x^{\frac{1}{2}}$.
$\frac{d}{dx}[x^{\frac{1}{2}}]$

Step:2.1.2

Apply the Power Rule, $\frac{d}{dx}[x^n]=n{x}^{n-1}$, where $n=\frac{1}{2}$.
$\frac{1}{2}{x}^{\frac{1}{2}-1}$

Step:2.1.3

Express $-1$ as a fraction with a common denominator by multiplying by $\frac{2}{2}$.
$\frac{1}{2}{x}^{\frac{1}{2}-1\cdot \frac{2}{2}}$

Step:2.1.4

Combine $-1$ and $\frac{2}{2}$.
$\frac{1}{2}{x}^{\frac{1}{2}-\frac{2}{2}}$

Step:2.1.5

Add the numerators over the common denominator.
$\frac{1}{2}{x}^{\frac{1-2}{2}}$

Step:2.1.6

Simplify the numerator.

Step:2.1.6.1

Multiply $-1$ by $2$.
$\frac{1}{2}{x}^{\frac{-1}{2}}$

Step:2.1.6.2

Subtract $2$ from $1$.
$\frac{1}{2}{x}^{-\frac{1}{2}}$

Step:2.1.7

Move the negative in front of the fraction.
$\frac{1}{2}{x}^{-\frac{1}{2}}$

Step:2.1.8

Simplify further.

Step:2.1.8.1

Apply the negative exponent rule $b^{-n}=\frac{1}{b^n}$.
$\frac{1}{2}\cdot \frac{1}{x^{\frac{1}{2}}}$

Step:2.1.8.2

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$.
$\frac{1}{2x^{\frac{1}{2}}}$

Step:2.2

Reformulate the integral with $u$ and $du$.
$\int 2\sin (u)du$

Step:3

Extract the constant $2$ from the integral as it is not dependent on $u$.
$2\int \sin (u)du$

Step:4

Integrate $\sin (u)$ with respect to $u$ to get $-\cos (u)$.
$2(-\cos (u)+C)$

Step:5

Simplify the expression.

Step:5.1

Simplify the result.
$2(-\cos (u))+C$

Step:5.2

Multiply $-1$ by $2$.
$-2\cos (u)+C$

Step:6

Substitute $u$ back with $x^{\frac{1}{2}}$.
$-2\cos (x^{\frac{1}{2}})+C$

Knowledge Notes:

1. Exponent Rules: These include the power of a power rule, product of powers rule, and negative exponent rule, which are fundamental in simplifying expressions with exponents.

2. u-Substitution: A technique used in integration to simplify the integral by substituting part of the integrand with a new variable, $u$. This often simplifies the integral into a more recognizable form.

3. Power Rule for Differentiation: A basic rule of differentiation that states if $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$.

4. Integration of Trigonometric Functions: The integral of $\sin (x)$ is $-\cos (x)$, and this is a basic integral that is often memorized.

5. Negative Exponent Rule: States that $a^{-n}=\frac{1}{a^n}$, which is used to transform expressions with negative exponents into fractions.

6. Constant Multiple Rule in Integration: Allows a constant to be moved outside of an integral, which can simplify the integration process.