Math  /  Calculus

QuestionEvaluate the integral 1911+x2dx\int_{1}^{\sqrt{9}} \frac{1}{1+x^{2}} d x

Studdy Solution

STEP 1

1. We are given the integral 1911+x2dx\int_{1}^{\sqrt{9}} \frac{1}{1+x^{2}} \, dx.
2. The function 11+x2\frac{1}{1+x^{2}} is a standard form that can be integrated using a known antiderivative.
3. The limits of integration are from 11 to 9\sqrt{9}.

STEP 2

1. Simplify the limits of integration.
2. Identify the antiderivative of the integrand 11+x2\frac{1}{1+x^{2}}.
3. Evaluate the antiderivative at the upper and lower limits.
4. Compute the definite integral by subtracting the evaluated lower limit from the evaluated upper limit.

STEP 3

Simplify the limits of integration:
The upper limit 9\sqrt{9} simplifies to 33.

STEP 4

Identify the antiderivative of 11+x2\frac{1}{1+x^{2}}:
The antiderivative of 11+x2\frac{1}{1+x^{2}} is arctan(x)\arctan(x).

STEP 5

Evaluate the antiderivative at the upper limit x=3x = 3:
arctan(3)\arctan(3).

STEP 6

Evaluate the antiderivative at the lower limit x=1x = 1:
arctan(1)\arctan(1).

STEP 7

Compute the definite integral:
Subtract the evaluated lower limit from the evaluated upper limit:
1311+x2dx=arctan(3)arctan(1)\int_{1}^{3} \frac{1}{1+x^{2}} \, dx = \arctan(3) - \arctan(1)
The value of the integral is:
arctan(3)π4\boxed{\arctan(3) - \frac{\pi}{4}}

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