The value of int_(1)^(sqrt3)(dx)/(1+x^(2)) is: (a) (pi)/(2) (b) (2pi)/(3) (c) (pi)/(6) (d) (pi)/(12)

Recognize standard form: Recognize the integral as a standard form. The integral of 1/(1+x^2) is a standard form that is recognized as the inverse tangent function, arctan(x). Therefore, the integral can be written as: ∫(dx)/(1+x^2) = arctan(x) + CApply limits of integration: Apply the limits of integration. We need to evaluate the definite integral from 1 to sqrt(3), which means we will substitute these values into the antiderivative we found in Step 1: ∫_(1)^(sqrt3)(dx)/(1+x^(2)) = arctan(sqrt(3)) - arctan(1)Evaluate arctan function: Evaluate the arctan function at the limits. arctan(sqrt(3)) corresponds to the angle whose tangent is sqrt(3), which is π/3 because tan(π/3) = sqrt(3). arctan(1) corresponds to the angle whose tangent is 1, which is π/4 because tan(π/4) = 1.Perform subtraction: Perform the subtraction to find the exact value. Now we subtract the two values: arctan(sqrt(3)) - arctan(1) = (π/3) - (π/4)Simplify expression: Simplify the expression. To subtract the fractions, find a common denominator, which is 12 in this case: (π/3) - (π/4) = (4π/12) - (3π/12) = (π/12)

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The value of $\newline$ $\int_{1}^{\sqrt{3}}\frac{dx}{1+x^{2}}$ is: $\newline$ (a) $\frac{\pi}{2}$ $\newline$ (b) $\frac{2\pi}{3}$ $\newline$ (c) $\frac{\pi}{6}$ $\newline$ (d) $\frac{\pi}{12}$

View step-by-step help

Home

Math Problems

Calculus

Evaluate definite integrals using the chain rule

Full solution

Q. The value of

\newline

\int_{1}^{\sqrt{3}}\frac{dx}{1+x^{2}}

is:

\newline

(a)

\frac{\pi}{2}

\newline

(b)

\frac{2\pi}{3}

\newline

(c)

\frac{\pi}{6}

\newline

(d)

\frac{\pi}{12}

Recognize standard form: Recognize the integral as a standard form. The integral of $\frac{1}{1+x^2}$ is a standard form that is recognized as the inverse tangent function, $\text{arctan}(x)$ . Therefore, the integral can be written as: $\int \frac{\mathrm{d}x}{1+x^2} = \text{arctan}(x) + C$
Apply limits of integration: Apply the limits of integration. We need to evaluate the definite integral from $1$ to $\sqrt{3}$ , which means we will substitute these values into the antiderivative we found in Step $1$ : $\int_{1}^{\sqrt{3}}\frac{dx}{1+x^{2}} = \arctan(\sqrt{3}) - \arctan(1)$
Evaluate arctan function: Evaluate the arctan function at the limits. $\newline$ $\arctan(\sqrt{3})$ corresponds to the angle whose tangent is $\sqrt{3}$ , which is $\frac{\pi}{3}$ because $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$ . $\newline$ $\arctan(1)$ corresponds to the angle whose tangent is $1$ , which is $\frac{\pi}{4}$ because $\tan\left(\frac{\pi}{4}\right) = 1$ .
Perform subtraction: Perform the subtraction to find the exact value. $\newline$ Now we subtract the two values: $\newline$ $\arctan(\sqrt{3}) - \arctan(1) = (\frac{\pi}{3}) - (\frac{\pi}{4})$
Simplify expression: Simplify the expression. $\newline$ To subtract the fractions, find a common denominator, which is $12$ in this case: $\newline$ $(\pi/3) - (\pi/4) = (4\pi/12) - (3\pi/12) = (\pi/12)$