Integrate Using u-Substitution integral of (5x^2-3)(2x) with respect to x
The given problem is a request to perform a definite or indefinite integration of a function using a technique called u-substitution. Specifically, the function to be integrated is (5x^2-3)(2x) with respect to the variable x. U-substitution is a method often used in calculus when an integral contains a composite function. It simplifies the integration process by substituting a part of the integrand with a new variable u, which is chosen to make the integral more manageable. The question implies finding the appropriate substitution, re-expressing the integral in terms of u, performing the integration, and then substituting back to the original variable x if it's an indefinite integral. The request is to explain what the question is asking, not to execute the integration itself.
$\int \left(\right. 5 x^{2} - 3 \left.\right) \left(\right. 2 x \left.\right) d x$
Answer
Solution:
Step 1:
Reposition the constant $2$ in front of the expression $(5x^2 - 3)$ to obtain $\int 2(5x^2 - 3)x \, dx$.
Step 2:
Set $u = 2(5x^2 - 3)$. Calculate $du = 20x \, dx$, which implies $\frac{1}{20}du = x \, dx$. Substitute $u$ and $du$ into the integral.
Step 2.1:
Define $u = 2(5x^2 - 3)$ and determine $\frac{du}{dx}$.
Step 2.1.1:
Differentiate $2(5x^2 - 3)$ with respect to $x$: $\frac{d}{dx}[2(5x^2 - 3)]$.
Step 2.1.2:
Apply the constant multiple rule to find the derivative: $2\frac{d}{dx}[5x^2 - 3]$.
Step 2.1.3:
Use the Sum Rule to differentiate $5x^2 - 3$: $2\left(\frac{d}{dx}[5x^2] + \frac{d}{dx}[-3]\right)$.
Step 2.1.4:
Differentiate $5x^2$ while treating $5$ as a constant: $2\left(5\frac{d}{dx}[x^2] + \frac{d}{dx}[-3]\right)$.
Step 2.1.5:
Apply the Power Rule, which states $\frac{d}{dx}[x^n] = nx^{n-1}$ for $n = 2$: $2\left(5(2x) + \frac{d}{dx}[-3]\right)$.
Step 2.1.6:
Multiply $2$ by $5$: $2(10x + \frac{d}{dx}[-3])$.
Step 2.1.7:
Since $-3$ is a constant, its derivative is $0$: $2(10x + 0)$.
Step 2.1.8:
Simplify the expression.
Step 2.1.8.1:
Combine $10x$ and $0$: $2(10x)$.
Step 2.1.8.2:
Multiply $10$ by $2$ to get $20x$.
Step 2.2:
Express the integral in terms of $u$ and $du$: $\int u \frac{1}{20} du$.
Step 3:
Combine $u$ and $\frac{1}{20}$ to form $\int \frac{u}{20} du$.
Step 4:
Extract the constant $\frac{1}{20}$ from the integral: $\frac{1}{20} \int u \, du$.
Step 5:
Integrate $u$ with respect to $u$ using the Power Rule to get $\frac{1}{2}u^2$: $\frac{1}{20}\left(\frac{1}{2}u^2 + C\right)$.
Step 6:
Simplify the expression.
Step 6.1:
Rewrite $\frac{1}{20}\left(\frac{1}{2}u^2 + C\right)$ as $\frac{1}{20} \cdot \frac{1}{2} u^2 + C$.
Step 6.2:
Combine the constants to get $\frac{1}{40}u^2 + C$.
Step 7:
Substitute back the original expression for $u$ to obtain $\frac{1}{40}(2(5x^2 - 3))^2 + C$.
Knowledge Notes:
u-Substitution: A technique used in integration that involves substituting part of the integral with a new variable $u$. This simplifies the integral, making it easier to solve.
Constant Multiple Rule: When taking the derivative of a constant multiplied by a function, the derivative is the constant multiplied by the derivative of the function.
Sum Rule: The derivative of a sum of two functions is the sum of the derivatives of those functions.
Power Rule: A basic rule for differentiation. If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Integration of Power Functions: The integral of $x^n$ with respect to $x$ is $\frac{1}{n+1}x^{n+1}$, provided $n \neq -1$.
Constants in Integration: Constants can be factored out of an integral. If $k$ is a constant, then $\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx$.
Simplification: The process of making an expression easier to understand or work with by combining like terms and using arithmetic operations.
Substituting Back: After integrating with respect to $u$, we substitute back the original expression that $u$ was set equal to, to return to the original variable.
