Integrate Using u-Substitution integral from 0 to 1 of 5/(x2+1) with respect to x

Question

Solvely Expert · Accepted 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: $5 \int_{0}^{1} \frac{1}{x^{2} + 1} d x$

Step 3:

Streamline the integrand.

Step 3.1:

Rearrange the denominator to $1 + x^{2}$ : $5 \int_{0}^{1} \frac{1}{1 + x^{2}} d x$

Step 3.2:

Express $1$ as $1^{2}$ to highlight the sum of squares: $5 \int_{0}^{1} \frac{1}{1^{2} + x^{2}} d x$

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 $\arctan (1)$ : $5 (\frac{π}{4} - \arctan (0))$

Step 6.1.2:

Find the exact value of $\arctan (0)$ : $5 (\frac{π}{4} - 0)$

Step 6.1.3:

Multiply $- 1$ by $0$ to maintain the expression: $5 (\frac{π}{4} + 0)$

Step 6.2:

Add $\frac{π}{4}$ and $0$ together: $5 \frac{π}{4}$

Step 6.3:

Combine the constant $5$ with $\frac{π}{4}$ : $\frac{5 π}{4}$

Step 7:

Present the result in various forms.

Exact Form: $\frac{5 π}{4}$

Decimal Form: $3.92699081 \dots$

Step 8:

Solution:"The integral of $\frac{5}{x^{2} + 1}$ from $0$ to $1$ is $\frac{5 π}{4}$ ."

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 $\frac{π}{4}$ because the tangent of $\frac{π}{4}$ 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 $π$ or other constants) or in decimal form for practical applications.

Integrate Using u-Substitution integral from 0 to 1 of 5/(x^2+1) with respect to x

Answer