Integrate Using u-Substitution integral of x^2 square root of x^3+33 with respect to x , u=x^3+33
The given problem is asking for the evaluation of an indefinite integral of a function involving a power of x and a square root of a polynomial expression in x. Specifically, the task is to find the antiderivative of x^2 times the square root of (x^3 + 33). The question hints at using the method of u-substitution, a common technique in calculus for simplifying integrals. U-substitution involves choosing a part of the integrand to substitute with a new variable u, such that the integral becomes easier to evaluate when expressed in terms of u. The chosen u in this case is suggested as u = x^3 + 33. This substitution is likely to simplify the integrand, as taking the derivative of u with respect to x will introduce a factor related to x^2, which is present in the original integrand. This could potentially transform the integral into a form involving u that is more straightforward to integrate.
$\int x^{2} \sqrt{x^{3} + 33} d x$,$u = x^{3} + 33$
Answer
Solution:
Step:1
Choose $u = x^{3} + 33$. Consequently, $d u = 3 x^{2} d x$. Express the integral in terms of $u$ and $d u$.
Step:1.1
Set $u = x^{3} + 33$ and compute $\frac{d u}{d x}$.
Step:1.1.1
Take the derivative of $x^{3} + 33$. $\frac{d}{d x} (x^{3} + 33)$
Step:1.1.2
Apply the Sum Rule to find the derivative of $x^{3} + 33$ with respect to $x$: $\frac{d}{d x} (x^{3}) + \frac{d}{d x} (33)$.
Step:1.1.3
Use the Power Rule, which states that the derivative of $x^{n}$ is $n x^{n - 1}$ for $n = 3$: $3 x^{2} + \frac{d}{d x} (33)$.
Step:1.1.4
Since the derivative of a constant is zero, the derivative of 33 with respect to $x$ is $0$: $3 x^{2} + 0$.
Step:1.1.5
Combine $3 x^{2}$ and $0$: $3 x^{2}$.
Step:1.2
Substitute $u$ and $d u$ into the integral: $\int \frac{\sqrt{u}}{3} d u$.
Step:2
Extract the constant $\frac{1}{3}$ from the integral: $\frac{1}{3} \int \sqrt{u} d u$.
Step:3
Express $\sqrt{u}$ as a power of $u$: $\frac{1}{3} \int u^{\frac{1}{2}} d u$.
Step:4
Integrate $u^{\frac{1}{2}}$ with respect to $u$ using the Power Rule: $\frac{1}{3} \left( \frac{2}{3} u^{\frac{3}{2}} + C \right)$.
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: $\frac{2}{9} \left( x^{3} + 33 \right)^{\frac{3}{2}} + C$.
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:
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 $d u$ is then the derivative of $u$ with respect to $x$ multiplied by $d x$.
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 $n x^{n-1}$.
Constant Multiple Rule: A constant can be factored out of the integral, which simplifies the integration process.
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.
Simplifying Expressions: After integration, it's common to simplify the expression by multiplying out constants and combining like terms.
Back Substitution: After integrating with respect to $u$, we substitute back in terms of the original variable $x$ to complete the problem.
