Integrate Using u-Substitution integral of (5x^2-3)(2x) with respect to x

Solution:

Step 1:

Reposition the constant $2$ in front of the expression $(5x^2-3)$ to obtain $\int 2(5x^2-3)xdx$.

Step 2:

Set $u=2(5x^2-3)$. Calculate $du=20xdx$, which implies $\frac{1}{20}du=xdx$. 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 $5x^2-3$: $2(\frac{d}{dx}[5x^2]+\frac{d}{dx}[-3])$.

Step 2.1.4:

Differentiate $5x^2$ while treating $5$ as a constant: $2(5\frac{d}{dx}[x^2]+\frac{d}{dx}[-3])$.

Step 2.1.5:

Apply the Power Rule, which states $\frac{d}{dx}[x^n]=n{x}^{n-1}$ for $n=2$: $2(5(2x)+\frac{d}{dx}[-3])$.

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: $\frac{1}{20}\int udu$.

Step 5:

Integrate $u$ with respect to $u$ using the Power Rule to get $\frac{1}{2}{u}^2$: $\frac{1}{20}(\frac{1}{2}{u}^2+C)$.

Step 6:

Simplify the expression.

Step 6.1:

Rewrite $\frac{1}{20}(\frac{1}{2}{u}^2+C)$ 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:

1. 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.

2. 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.

3. Sum Rule: The derivative of a sum of two functions is the sum of the derivatives of those functions.

4. Power Rule: A basic rule for differentiation. If $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$.

5. Integration of Power Functions: The integral of $x^n$ with respect to $x$ is $\frac{1}{n+1}{x}^{n+1}$, provided $n\ne -1$.

6. Constants in Integration: Constants can be factored out of an integral. If $k$ is a constant, then $\int k\cdot f(x)dx=k\cdot \int f(x)dx$.

7. Simplification: The process of making an expression easier to understand or work with by combining like terms and using arithmetic operations.

8. 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.