Integrate Using u-Substitution integral of 10x(5x^2-2)^2 with respect to x
This problem is asking you to perform integration on a function that is a product of a polynomial and a function raised to a power. The method suggested for solving this integral is u-substitution. This technique involves identifying a part of the integrand as 'u', which simplifies the expression, and then rewriting the entire integral in terms of 'u' before performing the integration. After integrating with respect to 'u', you would then substitute back in terms of the original variable 'x' to arrive at the solution.
$\int 10 x \left(\left(\right. 5 x^{2} - 2 \left.\right)\right)^{2} d x$
Answer
Solution:
Step 1
Assign $u = 5x^2 - 2$. Compute $du$ by differentiating $u$ with respect to $x$ and solve for $dx$.
Step 1.1
Set $u = 5x^2 - 2$ and calculate $\frac{du}{dx}$.
Step 1.1.1
Differentiate the expression $5x^2 - 2$ to find $\frac{d}{dx}(5x^2 - 2)$.
Step 1.1.2
Apply the Sum Rule to differentiate $5x^2 - 2$ as $\frac{d}{dx}(5x^2) + \frac{d}{dx}(-2)$.
Step 1.1.3
Compute the derivative of $5x^2$.
Step 1.1.3.1
With $5$ being a constant, the derivative of $5x^2$ is $5\frac{d}{dx}(x^2)$.
Step 1.1.3.2
Use the Power Rule, which states $\frac{d}{dx}(x^n) = nx^{n-1}$ for $n=2$, to find $5(2x)$.
Step 1.1.3.3
Multiply $5$ by $2$ to get $10x$.
Step 1.1.4
Apply the Constant Rule for differentiation.
Step 1.1.4.1
Since $-2$ is a constant, its derivative is $0$.
Step 1.1.4.2
Combine $10x$ and $0$ to obtain $10x$.
Step 1.2
Express the integral in terms of $u$ and $du$ as $\int u^2 du$.
Step 2
Integrate $u^2$ with respect to $u$ using the Power Rule to get $\frac{1}{3}u^3 + C$.
Step 3
Substitute back $5x^2 - 2$ for $u$ to obtain the final answer $\frac{1}{3}((5x^2 - 2))^3 + C$.
Knowledge Notes:
U-Substitution: This is a technique used in integration that involves substituting a part of the integrand with a new variable $u$. This is particularly useful when the integrand is a composite function.
Differentiation Rules:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives of those functions.
Constant Rule: The derivative of a constant is zero.
Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Integration Rules:
- Power Rule for Integration: If $f(x) = x^n$, then $\int f(x)dx = \frac{1}{n+1}x^{n+1} + C$, where $C$ is the constant of integration.
Substituting Back: After integrating with respect to $u$, it's necessary to substitute back the original variable to obtain the final result in terms of the original variable.
Constants in Integration: When integrating, a constant of integration ($C$) is added to represent the family of antiderivatives, since indefinite integration is only determined up to an additive constant.
