Math / Calculus

Question10) The improper integral $\int_{0}^{\infty} \frac{16 \tan ^{-1} x}{1+x^{2}} d x$ A) diverges B) converges to $2 \pi^{2}$ C) converges to $\frac{1}{2} \pi^{2}$ D) converges to $\frac{3}{2} \pi^{2}$ E) NOTA

Studdy Solution

STEP 1

1. The integral is improper because it has an infinite limit of integration.
2. We need to determine if the integral converges or diverges.
3. If it converges, we need to find the exact value.

STEP 2

1. Analyze the behavior of the integrand as $x \to \infty$ .
2. Evaluate the integral using a suitable technique.
3. Determine convergence or divergence.
4. Calculate the exact value if it converges.

STEP 3

First, analyze the behavior of the integrand $\frac{16 \tan^{-1} x}{1+x^2}$ as $x \to \infty$ .
As $x \to \infty$ , $\tan^{-1} x \to \frac{\pi}{2}$ . Therefore, the integrand behaves like:
$\frac{16 \cdot \frac{\pi}{2}}{1 + x^2} = \frac{8\pi}{1 + x^2}$
The function $\frac{8\pi}{1 + x^2}$ is similar to $\frac{1}{x^2}$ for large $x$ , which is known to be convergent.

STEP 4

Evaluate the integral using integration by parts. Let $u = \tan^{-1} x$ and $dv = \frac{16}{1 + x^2} dx$ .
Then $du = \frac{1}{1 + x^2} dx$ and $v = 16x$ .
Using integration by parts:
$\int u \, dv = uv - \int v \, du$
$= 16x \tan^{-1} x \Big|_0^\infty - \int_0^\infty 16x \cdot \frac{1}{1 + x^2} \, dx$

STEP 5

Evaluate the boundary terms:
As $x \to \infty$ , $16x \tan^{-1} x \to 16 \cdot \infty \cdot \frac{\pi}{2}$ , which diverges.
The boundary term at $x = 0$ is $16 \cdot 0 \cdot \tan^{-1}(0) = 0$ .
Since the boundary term diverges, the integral diverges.

STEP 6

Since the boundary term diverges, the integral does not converge to a finite value.
The integral diverges.
The integral diverges, so the correct answer is:
A) diverges

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Question10) The improper integral $\int_{0}^{\infty} \frac{16 \tan ^{-1} x}{1+x^{2}} d x$ A) diverges B) converges to $2 \pi^{2}$ C) converges to $\frac{1}{2} \pi^{2}$ D) converges to $\frac{3}{2} \pi^{2}$ E) NOTA

Studdy Solution

STEP 1

1. The integral is improper because it has an infinite limit of integration.
2. We need to determine if the integral converges or diverges.
3. If it converges, we need to find the exact value.

STEP 2

1. Analyze the behavior of the integrand as $x \to \infty$ .
2. Evaluate the integral using a suitable technique.
3. Determine convergence or divergence.
4. Calculate the exact value if it converges.

STEP 3

STEP 4

STEP 5

Evaluate the boundary terms:
As $x \to \infty$ , $16x \tan^{-1} x \to 16 \cdot \infty \cdot \frac{\pi}{2}$ , which diverges.
The boundary term at $x = 0$ is $16 \cdot 0 \cdot \tan^{-1}(0) = 0$ .
Since the boundary term diverges, the integral diverges.

STEP 6

Since the boundary term diverges, the integral does not converge to a finite value.
The integral diverges.
The integral diverges, so the correct answer is:
A) diverges

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Question10) The improper integral ∫0∞16tan⁡−1x1+x2dx\int_{0}^{\infty} \frac{16 \tan ^{-1} x}{1+x^{2}} d x∫0∞​1+x216tan−1x​dx A) diverges B) converges to 2π22 \pi^{2}2π2 C) converges to 12π2\frac{1}{2} \pi^{2}21​π2 D) converges to 32π2\frac{3}{2} \pi^{2}23​π2 E) NOTA

Studdy Solution

STEP 1

1. The integral is improper because it has an infinite limit of integration.2. We need to determine if the integral converges or diverges.3. If it converges, we need to find the exact value.

STEP 2

1. Analyze the behavior of the integrand as x→∞ x \to \infty x→∞.2. Evaluate the integral using a suitable technique.3. Determine convergence or divergence.4. Calculate the exact value if it converges.

STEP 3

STEP 4

STEP 5

STEP 6

Since the boundary term diverges, the integral does not converge to a finite value.The integral diverges. The integral diverges, so the correct answer is:A) diverges

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Question10) The improper integral $\int_{0}^{\infty} \frac{16 \tan ^{-1} x}{1+x^{2}} d x$ A) diverges B) converges to $2 \pi^{2}$ C) converges to $\frac{1}{2} \pi^{2}$ D) converges to $\frac{3}{2} \pi^{2}$ E) NOTA

1. The integral is improper because it has an infinite limit of integration.
2. We need to determine if the integral converges or diverges.
3. If it converges, we need to find the exact value.

1. Analyze the behavior of the integrand as $x \to \infty$ .
2. Evaluate the integral using a suitable technique.
3. Determine convergence or divergence.
4. Calculate the exact value if it converges.

Since the boundary term diverges, the integral does not converge to a finite value.
The integral diverges.
The integral diverges, so the correct answer is:
A) diverges