Integrate Using u-Substitution integral of x^2 square root of x^3+33 with respect to x , u=x^3+33

Solution:

Step:1

Choose $u=x^3+33$. Consequently, $du=3x^{2}dx$. Express the integral in terms of $u$ and $du$.

Step:1.1

Set $u=x^3+33$ and compute $\frac{du}{dx}$.

Step:1.1.1

Take the derivative of $x^3+33$. $\frac{d}{dx}(x^3+33)$

Step:1.1.2

Apply the Sum Rule to find the derivative of $x^3+33$ with respect to $x$: $\frac{d}{dx}(x^3)+\frac{d}{dx}(33)$.

Step:1.1.3

Use the Power Rule, which states that the derivative of $x^n$ is $n{x}^{n-1}$ for $n=3$: $3x^2+\frac{d}{dx}(33)$.

Step:1.1.4

Since the derivative of a constant is zero, the derivative of 33 with respect to $x$ is $0$: $3x^2+0$.

Step:1.1.5

Combine $3x^2$ and $0$: $3x^2$.

Step:1.2

Substitute $u$ and $du$ into the integral: $\int \frac{\sqrt{u}}{3}du$.

Step:2

Extract the constant $\frac{1}{3}$ from the integral: $\frac{1}{3}\int \sqrt{u}du$.

Step:3

Express $\sqrt{u}$ as a power of $u$: $\frac{1}{3}\int u^{\frac{1}{2}}du$.

Step:4

Integrate $u^{\frac{1}{2}}$ with respect to $u$ using the Power Rule: $\frac{1}{3}(\frac{2}{3}{u}^{\frac{3}{2}}+C)$.

Step:5

Simplify the expression.

Step:5.1

Rewrite the expression as a product: $\frac{1}{3}\cdot \frac{2}{3}{u}^{\frac{3}{2}}+C$.

Step:5.2

Multiply the constants: $\frac{2}{9}{u}^{\frac{3}{2}}+C$.

Step:6

Substitute back the original variable: $\frac{2}{9}{(x^3+33)}^{\frac{3}{2}}+C$.

Knowledge Notes:

The problem involves integration using u-substitution, a technique often used to simplify integrals by changing the variable of integration. Here are the relevant knowledge points:

1. u-Substitution: This is a method for evaluating integrals. By choosing $u$ as a function of $x$, we can transform the integral into a simpler form. The differential $du$ is then the derivative of $u$ with respect to $x$ multiplied by $dx$.

2. Derivative Rules: The Sum Rule states that the derivative of a sum is the sum of the derivatives. The Power Rule states that the derivative of $x^n$ is $n{x}^{n-1}$.

3. Constant Multiple Rule: A constant can be factored out of the integral, which simplifies the integration process.

4. Integration of Power Functions: When integrating power functions of the form $u^n$, the Power Rule for integration states that $\int u^{n}du=\frac{1}{n+1}{u}^{n+1}+C$, where $C$ is the constant of integration.

5. Simplifying Expressions: After integration, it's common to simplify the expression by multiplying out constants and combining like terms.

6. Back Substitution: After integrating with respect to $u$, we substitute back in terms of the original variable $x$ to complete the problem.