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 $\frac{d}{dx}[x^n] = nx^{n-1}$, 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: $\frac{d}{dx}(x^n) = nx^{n-1}$.

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

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