Integrate Using u-Substitution integral from 0 to 10 of 4x^2+7 with respect to x
The problem is asking for the application of a mathematical technique called u-substitution to solve a definite integral. U-substitution is a method used in calculus to simplify integrals by making a substitution of variables. The integral provided is the function 4x^2+7, and the integration is to be performed over the interval from x=0 to x=10. The task is to identify an appropriate substitution to simplify the integral and then use this substitution to find the definite integral of the function over the given range.
$\int_{0}^{10} 4 x^{2} + 7 d x$
Answer
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: $\int_{0}^{10} 4x^2 dx + \int_{0}^{10} 7 dx$.
Step 3:
Extract the constant $4$ from the first integral: $4 \int_{0}^{10} x^2 dx + \int_{0}^{10} 7 dx$.
Step 4:
Using the Power Rule, integrate $x^2$ with respect to $x$ to get $\frac{1}{3}x^3$: $4 \left[ \frac{1}{3}x^3 \right]_{0}^{10} + \int_{0}^{10} 7 dx$.
Step 5:
Combine the constants $\frac{1}{3}$ and $x^3$: $4 \left[ \frac{x^3}{3} \right]_{0}^{10} + \int_{0}^{10} 7 dx$.
Step 6:
Apply the rule for integrating a constant: $4 \left[ \frac{x^3}{3} \right]_{0}^{10} + 7x \bigg|_{0}^{10}$.
Step 7:
Perform the substitution and simplification.
Step 7.1:
Evaluate $\frac{x^3}{3}$ at $10$ and $0$: $4 \left( \frac{(10)^3}{3} - \frac{0^3}{3} \right) + 7x \bigg|_{0}^{10}$.
Step 7.2:
Evaluate $7x$ at $10$ and $0$: $4 \left( \frac{1000}{3} - \frac{0}{3} \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3:
Simplify the expression.
Step 7.3.1:
Calculate $10^3$: $4 \left( \frac{1000}{3} - \frac{0}{3} \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3.2:
Any number raised to the power of $0$ is $1$: $4 \left( \frac{1000}{3} - \frac{0}{3} \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3.3:
Eliminate the common factors.
Step 7.3.3.1:
Extract the factor of $3$ from $0$: $4 \left( \frac{1000}{3} - \frac{3(0)}{3} \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3.3.2:
Cancel out the common factors.
Step 7.3.3.2.1:
Extract the factor of $3$ from $3$: $4 \left( \frac{1000}{3} - \frac{3 \cdot 0}{3 \cdot 1} \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3.3.2.2:
Eliminate the common factor: $4 \left( \frac{1000}{3} - \frac{\cancel{3} \cdot 0}{\cancel{3} \cdot 1} \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3.3.2.3:
Rewrite the simplified expression: $4 \left( \frac{1000}{3} - \frac{0}{1} \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3.3.2.4:
Divide $0$ by $1$: $4 \left( \frac{1000}{3} - 0 \right) + 7 \cdot 10 - 7 \cdot 0$.
Step 7.3.4:
Multiply $-1$ by $0$: $4 \left( \frac{1000}{3} + 0 \right) + 7 \cdot 10 - 7 \cdot 0$.
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: $1403.\overline{3}$
- 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:
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.
Separation of Integrals: An integral of a sum can be separated into the sum of integrals of each term.
Constant Multiple Rule: Constants can be factored out of an integral.
Power Rule for Integration: The integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, provided $n \neq -1$.
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.
Simplifying Expressions: Combine like terms and simplify arithmetic operations to reach the final result.
