Integrate Using u-Substitution integral of 1/(y square root of y) with respect to y

Solution:

Step 1:

We cannot use u-substitution to solve this integral. An alternative approach will be used.

Step 2:

Begin by simplifying the integrand.

Step 2.1:

Start simplification.

Step 2.1.1:

Convert the square root to an exponent using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. Thus, $\sqrt{y}$ becomes $y^{\frac{1}{2}}$. The integral now is $\int \frac{1}{y \cdot y^{\frac{1}{2}}} dy$.

Step 2.1.2:

Combine the terms $y$ and $y^{\frac{1}{2}}$ by summing their exponents.

Step 2.1.2.1:

Combine $y$ and $y^{\frac{1}{2}}$.

Step 2.1.2.1.1:

Express $y$ as $y^1$. The integral becomes $\int \frac{1}{y^{1} y^{\frac{1}{2}}} dy$.

Step 2.1.2.1.2:

Apply the exponent rule $a^{m} a^{n} = a^{m + n}$ to combine the exponents. The integral simplifies to $\int \frac{1}{y^{1 + \frac{1}{2}}} dy$.

Step 2.1.2.2:

Express the number 1 as a fraction with a common denominator. The integral is now $\int \frac{1}{y^{\frac{2}{2} + \frac{1}{2}}} dy$.

Step 2.1.2.3:

Combine the numerators over the common denominator. The integral is $\int \frac{1}{y^{\frac{2 + 1}{2}}} dy$.

Step 2.1.2.4:

Add the numbers 2 and 1. The integral becomes $\int \frac{1}{y^{\frac{3}{2}}} dy$.

Step 2.2:

Now apply the basic rules of exponents.

Step 2.2.1:

Rewrite $y^{\frac{3}{2}}$ as a negative exponent by raising it to the power of $-1$. The integral is $\int (y^{\frac{3}{2}})^{-1} dy$.

Step 2.2.2:

Multiply the exponents in $(y^{\frac{3}{2}})^{-1}$.

Step 2.2.2.1:

Use the power rule for exponents, $(a^{m})^{n} = a^{mn}$. The integral becomes $\int y^{\frac{3}{2} \cdot -1} dy$.

Step 2.2.2.2:

Perform the multiplication of $\frac{3}{2} \cdot -1$.

Step 2.2.2.2.1:

Combine $\frac{3}{2}$ and $-1$. The integral is now $\int y^{\frac{3 \cdot -1}{2}} dy$.

Step 2.2.2.2.2:

Multiply 3 by $-1$. The integral simplifies to $\int y^{\frac{-3}{2}} dy$.

Step 2.2.2.3:

Bring the negative sign in front of the fraction. The integral is $\int y^{-\frac{3}{2}} dy$.

Step 3:

According to the Power Rule for integration, the integral of $y^{-\frac{3}{2}}$ with respect to $y$ is $-2y^{-\frac{1}{2}} + C$.

Step 4:

Finally, simplify the result.

Step 4.1:

Rewrite $-2y^{-\frac{1}{2}} + C$ as $-2\frac{1}{y^{\frac{1}{2}}} + C$.

Step 4.2:

Proceed with simplification.

Step 4.2.1:

Combine $-2$ and $\frac{1}{y^{\frac{1}{2}}}$. The result is $\frac{-2}{y^{\frac{1}{2}}} + C$.

Step 4.2.2:

Place the negative sign in front of the fraction to get the final answer: $-\frac{2}{y^{\frac{1}{2}}} + C$.

Knowledge Notes:

1. u-Substitution: A technique used in integration, which involves substituting a part of the integrand with a new variable 'u' to simplify the integral. It's not applicable in this case.

2. Simplifying the Integrand: Before integrating, it can be helpful to simplify the expression by rewriting square roots as exponents and combining like terms.

3. Exponent Rules: Important rules include $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$, $a^{m} a^{n} = a^{m + n}$, and $(a^{m})^{n} = a^{mn}$. These rules are used to simplify the integrand before integrating.

4. Power Rule for Integration: For any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$, where C is the constant of integration. This rule is used to find the antiderivative of the given function.

5. Negative Exponents: A term with a negative exponent, such as $x^{-n}$, can be rewritten as $\frac{1}{x^n}$. This is used to move terms from the denominator to the numerator or vice versa when simplifying expressions.

6. Constants of Integration: When performing indefinite integration, a constant of integration (denoted as 'C') is added to the result to account for all possible antiderivatives.