Integrate Using u-Substitution integral of 1/( square root of 5x+8) with respect to x

Solution:

Step 1

Assign $u=5x+8$. Consequently, $du=5dx$, which implies $\frac{1}{5}du=dx$. Express the integral in terms of $u$ and $du$.

Step 1.1

Set $u=5x+8$. Calculate $\frac{du}{dx}$.

Step 1.1.1

Differentiate $5x+8$. $\frac{d}{dx}[5x+8]$

Step 1.1.2

Utilize the Sum Rule to find the derivative of $5x+8$ with respect to $x$: $\frac{d}{dx}[5x]+\frac{d}{dx}[8]$.

Step 1.1.3

Compute $\frac{d}{dx}[5x]$.

Step 1.1.3.1

Given that $5$ is a constant, the derivative of $5x$ with respect to $x$ is $5\frac{d}{dx}[x]$.

Step 1.1.3.2

Apply the Power Rule, which states $\frac{d}{dx}[x^n]=n{x}^{n-1}$, where $n=1$: $5\cdot 1+\frac{d}{dx}[8]$.

Step 1.1.3.3

Multiply $5$ by $1$: $5+\frac{d}{dx}[8]$.

Step 1.1.4

Differentiate using the Constant Rule.

Step 1.1.4.1

Since $8$ is a constant, its derivative with respect to $x$ is $0$: $5+0$.

Step 1.1.4.2

Combine $5$ and $0$: $5$.

Step 1.2

Reformulate the integral using $u$ and $du$: $\int \frac{1}{\sqrt{u}}\cdot \frac{1}{5}du$.

Step 2

Simplify the integral.

Step 2.1

Combine $\frac{1}{\sqrt{u}}$ and $\frac{1}{5}$: $\int \frac{1}{5\sqrt{u}}du$.

Step 2.2

Rearrange to $\int \frac{1}{5\sqrt{u}}du$.

Step 3

Extract the constant $\frac{1}{5}$ from the integral: $\frac{1}{5}\int \frac{1}{\sqrt{u}}du$.

Step 4

Apply exponent rules.

Step 4.1

Express $\sqrt{u}$ as $u^{\frac{1}{2}}$: $\frac{1}{5}\int u^{-\frac{1}{2}}du$.

Step 4.2

Rewrite using negative exponent: $\frac{1}{5}\int (u^{\frac{1}{2}}{)}^{-1}du$.

Step 4.3

Simplify the exponent.

Step 4.3.1

Use the rule $(a^m{)}^n=a^{mn}$: $\frac{1}{5}\int u^{\frac{1}{2}\cdot -1}du$.

Step 4.3.2

Combine the exponents: $\frac{1}{5}\int u^{-\frac{1}{2}}du$.

Step 4.3.3

Place the negative sign in front: $\frac{1}{5}\int u^{-\frac{1}{2}}du$.

Step 5

Integrate $u^{-\frac{1}{2}}$ with respect to $u$ using the Power Rule: $\frac{1}{5}(2u^{\frac{1}{2}}+C)$.

Step 6

Finalize the simplification.

Step 6.1

Rewrite as $\frac{1}{5}\cdot 2u^{\frac{1}{2}}+C$.

Step 6.2

Simplify to $\frac{2}{5}{u}^{\frac{1}{2}}+C$.

Step 7

Substitute back $u$ with $5x+8$: $\frac{2}{5}(5x+8{)}^{\frac{1}{2}}+C$.

Knowledge Notes:

The problem involves integrating a function with respect to $x$ using the technique of $u$-substitution. This method is often used when an integral contains a composite function that can be simplified by substituting part of the integrand with a new variable $u$. The steps involve:

1. Choosing an appropriate substitution for $u$ that simplifies the integral.

2. Differentiating $u$ with respect to $x$ to find $du$ and expressing $dx$ in terms of $du$.

3. Rewriting the integral in terms of $u$ and $du$.

4. Simplifying the integral, if possible, by factoring out constants or applying algebraic manipulations.

5. Integrating with respect to $u$ using standard integration techniques, such as the Power Rule.

6. Simplifying the result and substituting back the original variable to find the antiderivative in terms of $x$.

Relevant rules and concepts used in this process include:

• The Sum Rule for differentiation: $\frac{d}{dx}(f(x)+g(x))=\frac{df}{dx}+\frac{dg}{dx}$.

• The Constant Rule for differentiation: $\frac{d}{dx}(c)=0$ for any constant $c$.

• The Power Rule for differentiation: $\frac{d}{dx}(x^n)=n{x}^{n-1}$.

• The Power Rule for integration: $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C$ for any $n\ne -1$.

• Basic rules of exponents, such as $a^{m\cdot n}=(a^m{)}^n$ and $a^{-n}=\frac{1}{a^n}$.