Integrate Using u-Substitution integral from 0 to 1 of 5/(x^2+1) with respect to x
The question asks for the application of the u-substitution technique to solve a definite integral. Specifically, you are required to find the area under the curve of the function 5/(x^2+1) from x=0 to x=1. U-substitution is a method used in calculus to simplify integration by replacing a segment of the integrand with a single variable u, hence simplifying the integral into a form that is easier to evaluate. The problem requires identifying an appropriate substitution, finding the derivative of that substitution to replace dx, changing the limits of integration to match the u values, and then performing the integration with respect to u. After integrating with respect to u, you then reverse the substitution to express the antiderivative in terms of x, and finally evaluate the definite integral using the new limits.
$\int_{0}^{1} \frac{5}{x^{2} + 1} d x$
Answer
Solution:
Step 1:
Identify that u-substitution is not applicable for this integral. We will use a different method to solve it.
Step 2:
Extract the constant $5$ from the integral to simplify the expression: $5 \int_{0}^{1} \frac{1}{x^2 + 1} dx$
Step 3:
Streamline the integrand.
Step 3.1:
Rearrange the denominator to $1 + x^2$: $5 \int_{0}^{1} \frac{1}{1 + x^2} dx$
Step 3.2:
Express $1$ as $1^2$ to highlight the sum of squares: $5 \int_{0}^{1} \frac{1}{1^2 + x^2} dx$
Step 4:
Recognize the integral as the antiderivative of $\arctan(x)$: $5 [\arctan(x)]_{0}^{1}$
Step 5:
Evaluate $\arctan(x)$ at the upper and lower limits of the integral: $5 (\arctan(1) - \arctan(0))$
Step 6:
Simplify the expression.
Step 6.1:
Break down each term.
Step 6.1.1:
Determine the exact value of $\arctan(1)$: $5 \left(\frac{\pi}{4} - \arctan(0)\right)$
Step 6.1.2:
Find the exact value of $\arctan(0)$: $5 \left(\frac{\pi}{4} - 0\right)$
Step 6.1.3:
Multiply $-1$ by $0$ to maintain the expression: $5 \left(\frac{\pi}{4} + 0\right)$
Step 6.2:
Add $\frac{\pi}{4}$ and $0$ together: $5 \frac{\pi}{4}$
Step 6.3:
Combine the constant $5$ with $\frac{\pi}{4}$: $\frac{5\pi}{4}$
Step 7:
Present the result in various forms.
Exact Form: $\frac{5\pi}{4}$
Decimal Form: $3.92699081\ldots$
Step 8:
Solution:"The integral of $\frac{5}{x^2+1}$ from $0$ to $1$ is $\frac{5\pi}{4}$."
Knowledge Notes:
U-substitution is a technique used in integration when the integrand is a product of a function and its derivative. In this case, it was not applicable.
Constants can be factored out of integrals, which simplifies the integration process.
The integral of $\frac{1}{1+x^2}$ with respect to $x$ is $\arctan(x)$, which is a standard result from trigonometric integrals.
Evaluating definite integrals involves finding the antiderivative at the upper limit of integration and subtracting the antiderivative at the lower limit.
The arctangent function, $\arctan(x)$, gives the angle whose tangent is $x$. It is particularly useful for integrating functions that resemble the derivative of $\arctan(x)$.
The exact value of $\arctan(1)$ is $\frac{\pi}{4}$ because the tangent of $\frac{\pi}{4}$ is $1$.
The exact value of $\arctan(0)$ is $0$ because the tangent of $0$ is $0$.
The result of an integral can be expressed in exact form (typically involving $\pi$ or other constants) or in decimal form for practical applications.
