Integrate Using u-Substitution integral from 0 to 10 of 4x^2+7 with respect to x

Solution:

Step 1:

We cannot use u-substitution for this integral. We will employ a different technique.

Step 2:

Decompose the integral into two separate integrals: ${\int }_0^{10}4x^{2}dx+{\int }_0^{10}7dx$.

Step 3:

Extract the constant $4$ from the first integral: $4{\int }_0^{10}{x}^{2}dx+{\int }_0^{10}7dx$.

Step 4:

Using the Power Rule, integrate $x^2$ with respect to $x$ to get $\frac{1}{3}{x}^3$: $4{\left[\frac{1}{3}{x}^3\right]}_0^{10}+{\int }_0^{10}7dx$.

Step 5:

Combine the constants $\frac{1}{3}$ and $x^3$: $4{\left[\frac{x^3}{3}\right]}_0^{10}+{\int }_0^{10}7dx$.

Step 6:

Apply the rule for integrating a constant: $4{\left[\frac{x^3}{3}\right]}_0^{10}+7x{|}_0^{10}$.

Step 7:

Perform the substitution and simplification.

Step 7.1:

Evaluate $\frac{x^3}{3}$ at $10$ and $0$: $4(\frac{(10{)}^3}{3}-\frac{0^3}{3})+7x{|}_0^{10}$.

Step 7.2:

Evaluate $7x$ at $10$ and $0$: $4(\frac{1000}{3}-\frac{0}{3})+7\cdot 10-7\cdot 0$.

Step 7.3:

Simplify the expression.

Step 7.3.1:

Calculate ${10}^3$: $4(\frac{1000}{3}-\frac{0}{3})+7\cdot 10-7\cdot 0$.

Step 7.3.2:

Any number raised to the power of $0$ is $1$: $4(\frac{1000}{3}-\frac{0}{3})+7\cdot 10-7\cdot 0$.

Step 7.3.3:

Eliminate the common factors.

Step 7.3.3.1:

Extract the factor of $3$ from $0$: $4(\frac{1000}{3}-\frac{3(0)}{3})+7\cdot 10-7\cdot 0$.

Step 7.3.3.2:

Cancel out the common factors.

Step 7.3.3.2.1:

Extract the factor of $3$ from $3$: $4(\frac{1000}{3}-\frac{3\cdot 0}{3\cdot 1})+7\cdot 10-7\cdot 0$.

Step 7.3.3.2.2:

Eliminate the common factor: $4(\frac{1000}{3}-\frac{\cancel{3}\cdot 0}{\cancel{3}\cdot 1})+7\cdot 10-7\cdot 0$.

Step 7.3.3.2.3:

Rewrite the simplified expression: $4(\frac{1000}{3}-\frac{0}{1})+7\cdot 10-7\cdot 0$.

Step 7.3.3.2.4:

Divide $0$ by $1$: $4(\frac{1000}{3}-0)+7\cdot 10-7\cdot 0$.

Step 7.3.4:

Multiply $-1$ by $0$: $4(\frac{1000}{3}+0)+7\cdot 10-7\cdot 0$.

Step 7.3.5:

Add $\frac{1000}{3}$ and $0$: $4\left(\frac{1000}{3}\right)+7\cdot 10-7\cdot 0$.

Step 7.3.6:

Combine $4$ and $\frac{1000}{3}$: $\frac{4\cdot 1000}{3}+7\cdot 10-7\cdot 0$.

Step 7.3.7:

Multiply $4$ by $1000$: $\frac{4000}{3}+7\cdot 10-7\cdot 0$.

Step 7.3.8:

Multiply $7$ by $10$: $\frac{4000}{3}+70-7\cdot 0$.

Step 7.3.9:

Multiply $-7$ by $0$: $\frac{4000}{3}+70+0$.

Step 7.3.10:

Add $70$ and $0$: $\frac{4000}{3}+70$.

Step 7.3.11:

Convert $70$ to a fraction with the same denominator: $\frac{4000}{3}+70\cdot \frac{3}{3}$.

Step 7.3.12:

Combine $70$ and $\frac{3}{3}$: $\frac{4000}{3}+\frac{70\cdot 3}{3}$.

Step 7.3.13:

Combine the numerators over the common denominator: $\frac{4000+70\cdot 3}{3}$.

Step 7.3.14:

Simplify the numerator.

Step 7.3.14.1:

Multiply $70$ by $3$: $\frac{4000+210}{3}$.

Step 7.3.14.2:

Add $4000$ and $210$: $\frac{4210}{3}$.

Step 8:

The final answer can be expressed in various forms:

• Exact Form: $\frac{4210}{3}$
• Decimal Form: $1403.\bar{3}$
• Mixed Number Form: $1403\frac{1}{3}$.

Step 9:

End of the solution process.

Solution:"The integral of the function $4x^2+7$ from $0$ to $10$ with respect to $x$ is $\frac{4210}{3}$ or $1403\frac{1}{3}$ in mixed number form."

Knowledge Notes:

1. U-Substitution: A technique used in integration, which involves substituting a part of the integrand with a new variable $u$ to simplify the integral. This method is not applicable here.

2. Separation of Integrals: An integral of a sum can be separated into the sum of integrals of each term.

3. Constant Multiple Rule: Constants can be factored out of an integral.

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

5. Evaluating Definite Integrals: To evaluate a definite integral, find the antiderivative, then subtract the value of the antiderivative at the lower limit from the value at the upper limit.

6. Simplifying Expressions: Combine like terms and simplify arithmetic operations to reach the final result.