Q. 1 Solve each of the following inde... [FREE SOLUTION]

Chapter 6: Q. 1 (page 569)

Solve each of the following indefinite integrals
a) $鈭� \frac{1}{y^{2}} d y$
b) $鈭� \frac{1}{1 - y} d y$
c) $鈭� \frac{1}{y^{2} + 1} d y$
d) $鈭� \frac{y}{y^{2} + 1} d y$
e) $鈭� \frac{1}{y (1 - y)} d y$
f) $鈭� \frac{1}{y (1 - 3 y)} d y$

Short Answer

Expert verified

The integral is:

$a) - \frac{1}{y} + c b) - \ln (1 - y) + c c) \tan^{- 1} y + c d) \frac{1}{2} \ln (y^{2} + 1) + c e) \ln (\frac{y}{1 - y}) + c f) \ln (\frac{y}{1 - 3 y}) + c$

Step by step solution

Part (a) Step 1. Given information

The integrals to be solved are:

$鈭� \frac{1}{y^{2}} d y, 鈭� \frac{1}{1 - y} d y, 鈭� \frac{1}{y^{2} + 1} d y 鈭� \frac{y}{y^{2} + 1} d y 鈭� \frac{1}{y (1 - y)} d y 鈭� \frac{1}{y (1 - 3 y)} d y$

Part (a) step 2. Solve integral:

$鈭� \frac{1}{y^{2}} d y = 鈭� y^{- 2} d y = \frac{y^{- 2 + 1}}{- 2 + 1} + c = \frac{y^{- 1}}{- 1} + c = - \frac{1}{y} + c$

Where c is a constant.

Part (b) step 1. Solve integral:

$鈭� \frac{1}{1 - y} d y$

Let $z = 1 - y d z = - d y$

so, $鈭� \frac{1}{1 - y} d y = 鈭� - \frac{1}{z} d z = - \ln z + c = \ln (1 - y) + c$

Part (c) step 1. Solve integral:

$鈭� \frac{1}{1 + y^{2}} d y = \tan^{- 1} y + c$

Part (d) step 1. Solve integral:

$鈭� \frac{y}{y^{2} + 1} d y$

let $z = y^{2} + 1 d z = 2 y d y y d y = \frac{1}{2} d z$

So,

$鈭� \frac{y}{y^{2} + 1} d y = 鈭� \frac{d z}{2 z} = \frac{1}{2} \ln z + c = \frac{1}{2} \ln (1 + y^{2}) + c$

Part (e) step 1. Solve integral:

$鈭� \frac{1}{y (1 - y)} d y = 鈭� (\frac{1}{y} + \frac{1}{1 - y}) d y = 鈭� \frac{1}{y} d y + 鈭� \frac{1}{1 - y} d y = \ln y - \ln (1 - y) + c = \ln (\frac{y}{1 - y}) + c$

Part (f) step 1. Solve integral:

$鈭� \frac{1}{y (1 - 3 y)} d y = 鈭� (\frac{1}{y} + \frac{3}{1 - 3 y}) d y = 鈭� \frac{1}{y} d y + 鈭� \frac{3}{1 - 3 y} d y = \ln y + 鈭� \frac{3}{1 - 3 y} d y$

Let $1 - 3 y = z - 3 d y = d z 3 d y = - d z$

So, $\ln y + 鈭� \frac{3}{1 - 3 y} d y = \ln y - 鈭� \frac{d z}{z} = \ln y - \ln z + c = \ln y - \ln (1 - 3 y) + c = \ln (\frac{y}{1 - 3 y}) + c$

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Solve each of the following indefinite integrals
a) $鈭� \frac{1}{y^{2}} d y$
b) $鈭� \frac{1}{1 - y} d y$
c) $鈭� \frac{1}{y^{2} + 1} d y$
d) $鈭� \frac{y}{y^{2} + 1} d y$
e) $鈭� \frac{1}{y (1 - y)} d y$
f) $鈭� \frac{1}{y (1 - 3 y)} d y$

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) step 2. Solve integral:

Part (b) step 1. Solve integral:

Part (c) step 1. Solve integral:

Part (d) step 1. Solve integral:

Part (e) step 1. Solve integral:

Part (f) step 1. Solve integral:

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91影视

Solve each of the following indefinite integralsa)鈭�1y2dyb)鈭�11-ydyc)鈭�1y2+1dyd)鈭�yy2+1dye)鈭�1y(1-y)dyf)鈭�1y(1-3y)dy

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) step 2. Solve integral:

Part (b) step 1. Solve integral:

Part (c) step 1. Solve integral:

Part (d) step 1. Solve integral:

Part (e) step 1. Solve integral:

Part (f) step 1. Solve integral:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Probability and Statistics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Solve each of the following indefinite integrals
a) $鈭� \frac{1}{y^{2}} d y$
b) $鈭� \frac{1}{1 - y} d y$
c) $鈭� \frac{1}{y^{2} + 1} d y$
d) $鈭� \frac{y}{y^{2} + 1} d y$
e) $鈭� \frac{1}{y (1 - y)} d y$
f) $鈭� \frac{1}{y (1 - 3 y)} d y$