Math / Calculus

Question $\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right)$

Studdy Solution

STEP 1

1. We are given the integral: $\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right) \, dx$ .
2. We need to find the indefinite integral of the given expression.

STEP 2

1. Break down the integral into separate terms.
2. Integrate each term individually.
3. Combine the results to obtain the final solution.

STEP 3

Break down the integral into separate terms:
$\int \left( \frac{2x}{x^2+1} \right) \, dx + \int \left( \frac{1}{x^2+1} \right) \, dx - \int \left( \frac{2}{x-1} \right) \, dx + \int \left( \frac{1}{(x-1)^2} \right) \, dx$

STEP 4

Integrate the first term: $\int \left( \frac{2x}{x^2+1} \right) \, dx$ .
Use the substitution $u = x^2 + 1$ , $du = 2x \, dx$ .
$\int \frac{2x}{x^2+1} \, dx = \int \frac{1}{u} \, du = \ln |u| + C_1 = \ln |x^2 + 1| + C_1$

STEP 5

Integrate the second term: $\int \left( \frac{1}{x^2+1} \right) \, dx$ .
This is a standard integral:
$\int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C_2$

STEP 6

Integrate the third term: $-\int \left( \frac{2}{x-1} \right) \, dx$ .
This is a standard integral:
$-\int \frac{2}{x-1} \, dx = -2 \ln |x-1| + C_3$

STEP 7

Integrate the fourth term: $\int \left( \frac{1}{(x-1)^2} \right) \, dx$ .
This is a standard integral:
$\int \frac{1}{(x-1)^2} \, dx = -\frac{1}{x-1} + C_4$

STEP 8

Combine the results:
$\int \left(\frac{2x}{x^2+1} + \frac{1}{x^2+1} - \frac{2}{x-1} + \frac{1}{(x-1)^2}\right) \, dx = \ln |x^2 + 1| + \tan^{-1}(x) - 2 \ln |x-1| - \frac{1}{x-1} + C$
where $C = C_1 + C_2 + C_3 + C_4$ is the constant of integration.
The solution to the integral is:
$\boxed{\ln |x^2 + 1| + \tan^{-1}(x) - 2 \ln |x-1| - \frac{1}{x-1} + C}$

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Question $\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right)$

Studdy Solution

STEP 1

1. We are given the integral: $\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right) \, dx$ .
2. We need to find the indefinite integral of the given expression.

STEP 2

1. Break down the integral into separate terms.
2. Integrate each term individually.
3. Combine the results to obtain the final solution.

STEP 3

Break down the integral into separate terms:
$\int \left( \frac{2x}{x^2+1} \right) \, dx + \int \left( \frac{1}{x^2+1} \right) \, dx - \int \left( \frac{2}{x-1} \right) \, dx + \int \left( \frac{1}{(x-1)^2} \right) \, dx$

STEP 4

Integrate the first term: $\int \left( \frac{2x}{x^2+1} \right) \, dx$ .
Use the substitution $u = x^2 + 1$ , $du = 2x \, dx$ .
$\int \frac{2x}{x^2+1} \, dx = \int \frac{1}{u} \, du = \ln |u| + C_1 = \ln |x^2 + 1| + C_1$

STEP 5

Integrate the second term: $\int \left( \frac{1}{x^2+1} \right) \, dx$ .
This is a standard integral:
$\int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C_2$

STEP 6

Integrate the third term: $-\int \left( \frac{2}{x-1} \right) \, dx$ .
This is a standard integral:
$-\int \frac{2}{x-1} \, dx = -2 \ln |x-1| + C_3$

STEP 7

Integrate the fourth term: $\int \left( \frac{1}{(x-1)^2} \right) \, dx$ .
This is a standard integral:
$\int \frac{1}{(x-1)^2} \, dx = -\frac{1}{x-1} + C_4$

STEP 8

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Question∫(2xx2+1+1x2+1−2x−1+1(x−1)2)\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right)∫(x2+12x​+x2+11​−x−12​+(x−1)21​)

Studdy Solution

STEP 1

1. We are given the integral: ∫(2xx2+1+1x2+1−2x−1+1(x−1)2) dx\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right) \, dx∫(x2+12x​+x2+11​−x−12​+(x−1)21​)dx.2. We need to find the indefinite integral of the given expression.

STEP 2

1. Break down the integral into separate terms.2. Integrate each term individually.3. Combine the results to obtain the final solution.

STEP 3

STEP 4

STEP 5

Integrate the second term: ∫(1x2+1) dx\int \left( \frac{1}{x^2+1} \right) \, dx∫(x2+11​)dx.This is a standard integral:∫1x2+1 dx=tan⁡−1(x)+C2\int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C_2∫x2+11​dx=tan−1(x)+C2​

STEP 6

Integrate the third term: −∫(2x−1) dx-\int \left( \frac{2}{x-1} \right) \, dx−∫(x−12​)dx.This is a standard integral:−∫2x−1 dx=−2ln⁡∣x−1∣+C3-\int \frac{2}{x-1} \, dx = -2 \ln |x-1| + C_3−∫x−12​dx=−2ln∣x−1∣+C3​

STEP 7

Integrate the fourth term: ∫(1(x−1)2) dx\int \left( \frac{1}{(x-1)^2} \right) \, dx∫((x−1)21​)dx.This is a standard integral:∫1(x−1)2 dx=−1x−1+C4\int \frac{1}{(x-1)^2} \, dx = -\frac{1}{x-1} + C_4∫(x−1)21​dx=−x−11​+C4​

STEP 8

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Start learning now

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Question $\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right)$

1. We are given the integral: $\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right) \, dx$ .
2. We need to find the indefinite integral of the given expression.

1. Break down the integral into separate terms.
2. Integrate each term individually.
3. Combine the results to obtain the final solution.

Integrate the second term: $\int \left( \frac{1}{x^2+1} \right) \, dx$ .
This is a standard integral:
$\int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C_2$

Integrate the third term: $-\int \left( \frac{2}{x-1} \right) \, dx$ .
This is a standard integral:
$-\int \frac{2}{x-1} \, dx = -2 \ln |x-1| + C_3$

Integrate the fourth term: $\int \left( \frac{1}{(x-1)^2} \right) \, dx$ .
This is a standard integral:
$\int \frac{1}{(x-1)^2} \, dx = -\frac{1}{x-1} + C_4$