Solved: Integrate 3//(1+x^2) for a limit [0,1]

Identify Integral: Identify the integral that needs to be solved. We need to integrate the function 3/(1+x^2) with respect to x over the interval [0,1].Recognize Standard Form: Recognize the standard integral form. The integral of 1/(1+x^2) with respect to x is a standard integral that equals arctan(x) or tan^(-1)(x).Apply Constant Multiple Rule: Apply the constant multiple rule. Since we have a constant multiple (3) outside the integral, we can pull it out in front of the integral sign. ∫3/(1+x^2) dx = 3 * ∫1/(1+x^2) dxIntegrate Using Standard Integral: Integrate using the standard integral. Now we integrate 1/(1+x^2) which is arctan(x). ∫1/(1+x^2) dx = arctan(x)Apply Constant to Antiderivative: Apply the constant to the antiderivative. Multiplying the antiderivative by the constant 3, we get: 3 * arctan(x)Evaluate Antiderivative Bounds: Evaluate the antiderivative from 0 to 1. We need to evaluate 3 * arctan(x) at x=1 and x=0 and then subtract the latter from the former. 3 * arctan(1) - 3 * arctan(0)Calculate Arctan Values: Calculate the values of arctan at the bounds. arctan(1) = π/4 (since tan(π/4) = 1) arctan(0) = 0 (since tan(0) = 0)Substitute Values: Substitute the values into the expression. 3 * (π/4) - 3 * 0 = 3π/4 - 0 = 3π/4Write Final Answer: Write the final answer. The integral of 3/(1+x^2) from 0 to 1 is 3π/4.

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Integrate 3//(1+x^2) for a limit [0,1]

Solved: $\newline$ Integrate $\frac{3}{1+x^2}$ for a limit $[0,1]$

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Grade 8

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Q. Solved:

\newline

Integrate

\frac{3}{1+x^2}

for a limit

[0,1]

Identify Integral: Identify the integral that needs to be solved. $\newline$ We need to integrate the function $\frac{3}{1+x^2}$ with respect to $x$ over the interval $[0,1]$ .
Recognize Standard Form: Recognize the standard integral form. $\newline$ The integral of $\frac{1}{1+x^2}$ with respect to $x$ is a standard integral that equals $\arctan(x)$ or $\tan^{-1}(x)$ .
Apply Constant Multiple Rule: Apply the constant multiple rule. Since we have a constant multiple $3$ outside the integral, we can pull it out in front of the integral sign. $\int \frac{3}{1+x^2} \, dx = 3 \times \int \frac{1}{1+x^2} \, dx$
Integrate Using Standard Integral: Integrate using the standard integral. $\newline$ Now we integrate $1/(1+x^2)$ which is $\arctan(x)$ . $\newline$ $\int \frac{1}{1+x^2} \, dx = \arctan(x)$
Apply Constant to Antiderivative: Apply the constant to the antiderivative. $\newline$ Multiplying the antiderivative by the constant $3$ , we get: $\newline$ $3 \times \text{arctan}(x)$
Evaluate Antiderivative Bounds: Evaluate the antiderivative from $0$ to $1$ . We need to evaluate $3 \times \arctan(x)$ at $x=1$ and $x=0$ and then subtract the latter from the former. $3 \times \arctan(1) - 3 \times \arctan(0)$
Calculate Arctan Values: Calculate the values of arctan at the bounds. $\newline$ arctan( $1$ ) = $\frac{\pi}{4}$ (since $\tan(\frac{\pi}{4}) = 1$ ) $\newline$ arctan( $0$ ) = $0$ (since $\tan(0) = 0$ )
Substitute Values: Substitute the values into the expression. $\newline$ $3 \times (\pi/4) - 3 \times 0 = \frac{3\pi}{4} - 0 = \frac{3\pi}{4}$
Write Final Answer: Write the final answer. $\newline$ The integral of $\frac{3}{1+x^2}$ from $0$ to $1$ is $\frac{3\pi}{4}$ .