Integrate Using u-Substitution integral of 10x(5x^2-2)^2 with respect to x

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 $\frac{d}{dx}(x^n)=n{x}^{n-1}$ 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:

1. 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.

2. 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 $f^{\prime }(x)=n{x}^{n-1}$.

3. 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.

4. 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.

5. 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.