Maharashtra State BoardHSC Commerce 12th Board Exam
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Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 5 - Integration [Latest edition]

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Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board - Shaalaa.com
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Chapter 5: Integration

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6Miscellaneous Exercise 5
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Exercise 5.1 [Page 119]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 IntegrationExercise 5.1 [Page 119]

Exercise 5.1 | Q 1 | Page 119

Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx

Exercise 5.1 | Q 2 | Page 119

Evaluate `int (1 + "x" + "x"^2/(2!))`dx

Exercise 5.1 | Q 3 | Page 119

Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx

Exercise 5.1 | Q 4 | Page 119

Evaluate `int (3"x"^2 - 5)^2` dx

Exercise 5.1 | Q 5 | Page 119

Evaluate `int 1/("x" ("x - 1"))` dx

Exercise 5.1 | Q 6 | Page 119

If f '(x) = x2 + 5 and f(0) = - 1, then find the value of f(x).

Exercise 5.1 | Q 7 | Page 119

If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).

Exercise 5.1 | Q 8 | Page 119

If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).

Exercise 5.2 [Pages 122 - 123]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 IntegrationExercise 5.2 [Pages 122 - 123]

Exercise 5.2 | Q 1 | Page 122

Evaluate the following.

`int "x" sqrt(1 + "x"^2)` dx

Exercise 5.2 | Q 2 | Page 123

Evaluate the following.

`int "x"^3/sqrt(1 + "x"^4)` dx

Exercise 5.2 | Q 3 | Page 123

Evaluate the following.

`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx

Exercise 5.2 | Q 4 | Page 123

Evaluate the following.

`int (1 + "x")/("x" + "e"^"-x")` dx

Exercise 5.2 | Q 5 | Page 123

Evaluate the following.

∫ (x + 1)(x + 2)7 (x + 3)dx

Exercise 5.2 | Q 6 | Page 123

Evaluate the following.

`int 1/("x" log "x")`dx

Exercise 5.2 | Q 7 | Page 123

Evaluate the following.

`int "x"^5/("x"^2 + 1)`dx

Exercise 5.2 | Q 8 | Page 123

Evaluate the following.

`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx

Exercise 5.2 | Q 9 | Page 123

Evaluate the following.

`int 1/(sqrt"x" + "x")` dx

Exercise 5.2 | Q 10 | Page 123

Evaluate the following.

`int 1/("x"("x"^6 + 1))` dx

Exercise 5.3 [Page 123]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 IntegrationExercise 5.3 [Page 123]

Exercise 5.3 | Q 1 | Page 123

Evaluate the following.

`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt

Exercise 5.3 | Q 2 | Page 123

Evaluate the following.

`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx

Exercise 5.3 | Q 3 | Page 123

Evaluate the following.

`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx

Exercise 5.3 | Q 4 | Page 123

Evaluate the following.

`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx

Exercise 5.4 [Pages 128 - 129]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 IntegrationExercise 5.4 [Pages 128 - 129]

Exercise 5.4 | Q 1 | Page 129

Evaluate the following.

`int 1/(4"x"^2 - 1)` dx

Exercise 5.4 | Q 2 | Page 129

Evaluate the following.

`int 1/("x"^2 + 4"x" - 5)` dx

Exercise 5.4 | Q 3 | Page 129

Evaluate the following.

`int 1/(4"x"^2 - 20"x" + 17)` dx

Exercise 5.4 | Q 4 | Page 129

Evaluate the following.

`int "x"/(4"x"^2 - 2"x"^2 - 3)` dx

Exercise 5.4 | Q 5 | Page 128

Evaluate the following.

`int "x"^3/(16"x"^8 - 25)` dx

Exercise 5.4 | Q 6 | Page 129

Evaluate the following.

`int 1/("a"^2 - "b"^2 "x"^2)` dx

Exercise 5.4 | Q 7 | Page 129

Evaluate the following.

`int 1/(7 + 6"x" - "x"^2)` dx

Exercise 5.4 | Q 8 | Page 129

Evaluate the following.

`int 1/(sqrt(3"x"^2 + 8))` dx

Exercise 5.4 | Q 9 | Page 129

Evaluate the following.

`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx

Exercise 5.4 | Q 10 | Page 129

Evaluate the following.

`int 1/(sqrt(3"x"^2 - 5))` dx

Exercise 5.4 | Q 11 | Page 129

Evaluate the following.

`int 1/(sqrt("x"^2 -8"x" - 20))` dx

Exercise 5.5 [Page 133]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 IntegrationExercise 5.5 [Page 133]

Exercise 5.5 | Q 1 | Page 133

Evaluate the following.

∫ x log x dx

Exercise 5.5 | Q 2 | Page 133

Evaluate the following.

`int "x"^2 "e"^"4x"`dx

Exercise 5.5 | Q 3 | Page 133

Evaluate the following.

`int "x"^2 "e"^"3x"`dx

Exercise 5.5 | Q 4 | Page 133

Evaluate the following.

`int "x"^3 "e"^("x"^2)`dx

Exercise 5.5 | Q 5 | Page 133

Evaluate the following.

`int "e"^"x" (1/"x" - 1/"x"^2)`dx

Exercise 5.5 | Q 6 | Page 133

Evaluate the following.

`int "e"^"x" "x"/("x + 1")^2` dx

Exercise 5.5 | Q 7 | Page 133

Evaluate the following.

`int "e"^"x" "x - 1"/("x + 1")^3` dx

Exercise 5.5 | Q 8 | Page 133

Evaluate the following.

`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx

Exercise 5.5 | Q 9 | Page 133

Evaluate the following.

`int [1/(log "x") - 1/(log "x")^2]` dx

Exercise 5.5 | Q 10 | Page 133

Evaluate the following.

`int (log "x")/(1 + log "x")^2` dx

Exercise 5.6 [Page 135]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 IntegrationExercise 5.6 [Page 135]

Exercise 5.6 | Q 1 | Page 135

Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx

Exercise 5.6 | Q 2 | Page 135

Evaluate: `int (2"x" + 1)/("x"("x - 1")("x - 4"))` dx

Exercise 5.6 | Q 3 | Page 135

Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx

Exercise 5.6 | Q 4 | Page 135

Evaluate: `int "x"/(("x - 1")^2("x + 2"))` dx

Exercise 5.6 | Q 5 | Page 135

Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx

Exercise 5.6 | Q 6 | Page 135

Evaluate: `int 1/("x"("x"^5 + 1))` dx

Exercise 5.6 | Q 7 | Page 135

Evaluate: `int 1/("x"("x"^"n" + 1))` dx

Exercise 5.6 | Q 8 | Page 135

Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx

Miscellaneous Exercise 5 [Pages 137 - 139]

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 IntegrationMiscellaneous Exercise 5 [Pages 137 - 139]

Miscellaneous Exercise 5 | Q 1.01 | Page 137

Choose the correct alternative from the following.

The value of `int "dx"/sqrt"1 - x"` is

  • `2sqrt(1 - "x") + "c"`

  • - `2sqrt(1 - "x") + "c"`

  • `sqrt"x"` + c

  • x + c

Miscellaneous Exercise 5 | Q 1.02 | Page 137

Choose the correct alternative from the following.

`int sqrt(1 - "x"^2) "dx"` =

  • `"x"/2 sqrt(1 + "x"^2) + 1/2 log ("x" + sqrt(1 + "x"^2))`+c

  • `2/3 (1 + "x"^2)^(3/2) + "c"`

  • `1/3 (1 + "x"^2)` + c

  • `("x")/sqrt(1 + "x"^2)` + c

Miscellaneous Exercise 5 | Q 1.03 | Page 137

Choose the correct alternative from the following.

`int "x"^2 (3)^("x"^3) "dx"` =

  • `(3)^("x"^3) + "c"`

  • `(3)^("x"^3)/(3 * log 3) + "c"`

  • `log 3 (3)^("x"^3)` + c

  • `"x"^2 (3)^("x"^3) + "c"`

Miscellaneous Exercise 5 | Q 1.04 | Page 137

Choose the correct alternative from the following.

`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?

  • `1/3`

  • `1/2`

  • `1/4`

  • 2

Miscellaneous Exercise 5 | Q 1.05 | Page 137

Choose the correct alternative from the following.

`int "dx"/(("x" - "x"^2))`= 

  • log x – log (1 – x) + c

  • log (1 - x2) + c

  • - log x + log(1 - x) + c

  • log (x - x2) + c 

Miscellaneous Exercise 5 | Q 1.06 | Page 137

Choose the correct alternative from the following.

`int "dx"/(("x - 8")("x + 7"))`= 

  • `1/15 log |("x" + 2)/("x - 1")|` + c

  • `1/15 log |("x + 8")/("x + 7")|` + c

  • `1/15 log |("x - 8")/("x + 7")|` + c

  • (x - 8)(x - 7) + c

Miscellaneous Exercise 5 | Q 1.07 | Page 137

Choose the correct alternative from the following.

`int ("x" + 1/"x")^3 "dx"` = 

  • `1/4 ("x" + 1/"x")^4` + c

  • `"x"^4/4 + "3x"^2/2 + 3 log "x" - 1/"2x"^2 + "c"`

  • `"x"^4/4 + "3x"^2/2 + 3 log "x" + 1/"x"^2 + "c"`

  • `("x" - "x"^-1)^3` + c

Miscellaneous Exercise 5 | Q 1.08 | Page 137

Choose the correct alternative from the following.

`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` = 

  • `"e"^"x" - 1/(3"e"^"3x")` + c

  • `"e"^"x" + 1/(3"e"^"3x")` + c

  • `"e"^"-x" + 1/(3"e"^"3x")` + c

  • `"e"^"-x" + 1/(3"e"^"3x") + "c"`

Miscellaneous Exercise 5 | Q 1.09 | Page 137

Choose the correct alternative from the following.

`int (1 - "x")^(-2) "dx"` = 

  • `(1 + "x")^-1` + c

  • `(1 - "x")^-1` + c

  • `(1 - "x")^-1 - 1` + c

  • `(1 - "x")^-1 + 1` + c

Miscellaneous Exercise 5 | Q 1.1 | Page 138

Choose the correct alternative from the following.

`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5  "dx"` = 

  • `(-1)/"x + 1"` + c

  • `((-1)/"x + 1")^5` + c

  • log(x + 1) + c

  • log |x + 1|5 + c 

Miscellaneous Exercise 5 | Q 2.1 | Page 138

Fill in the Blank.

`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x5 + ______ x3 + 5x + c

Miscellaneous Exercise 5 | Q 2.2 | Page 138

Fill in the Blank.

`int ("x"^2 + "x" - 6)/(("x - 2")("x - 1")) "dx" = "x"` + ______ + c

Miscellaneous Exercise 5 | Q 2.3 | Page 138

Fill in the Blank.

If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______

Miscellaneous Exercise 5 | Q 2.4 | Page 138

Fill in the Blank.

To find the value of `int ((1 + log "x") "dx")/"x"` the proper substitution is ________

Miscellaneous Exercise 5 | Q 2.5 | Page 138

Fill in the Blank.

`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______

Miscellaneous Exercise 5 | Q 3.1 | Page 138

State whether the following statement is True or False.

The proper substitution for `int x(x^x)^x (2log x + 1)  "d"x` is `(x^x)^x` = t

  • True

  • False

Miscellaneous Exercise 5 | Q 3.2 | Page 138

State whether the following statement is True or False.

If `int x  "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.

  • True

  • False

Miscellaneous Exercise 5 | Q 3.3 | Page 138

State whether the following statement is True or False.

If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`

  • True

  • False

Miscellaneous Exercise 5 | Q 3.4 | Page 138

State whether the following statement is True or False.

If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.

  • True

  • False

Miscellaneous Exercise 5 | Q 3.5 | Page 138

State whether the following statement is True or False.

For `int ("x - 1")/("x + 1")^3  "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.

  • True

  • False

Miscellaneous Exercise 5 | Q 4.1 | Page 138

Evaluate `int (5"x"^2 - 6"x" + 3)/("2x" - 3)` dx

Miscellaneous Exercise 5 | Q 4.1 | Page 138

Evaluate `int (5"x" + 1)^(4/9)` dx

Miscellaneous Exercise 5 | Q 4.1 | Page 138

Evaluate `int 1/((2"x" + 3))` dx

Miscellaneous Exercise 5 | Q 4.1 | Page 138

Evaluate `int "x - 1"/sqrt("x + 4")` dx

Miscellaneous Exercise 5 | Q 4.1 | Page 138

Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).

Miscellaneous Exercise 5 | Q 4.1 | Page 138

Evaluate: ∫ |x| dx if x < 0

Miscellaneous Exercise 5 | Q 4.2 | Page 138

Evaluate: Find the primitive of `1/(1 + "e"^"x")`

Miscellaneous Exercise 5 | Q 4.2 | Page 138

Evaluate: `int ("ae"^"x" + "be"^-"x")/(("ae"^"x" - "be"^-"x"))` dx

Miscellaneous Exercise 5 | Q 4.2 | Page 138

Evaluate: `int 1/(2"x" + 3"x" log"x")` dx

Miscellaneous Exercise 5 | Q 4.2 | Page 138

Evaluate: `int 1/(sqrt("x") + "x")` dx

Miscellaneous Exercise 5 | Q 4.2 | Page 138

Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx

Miscellaneous Exercise 5 | Q 4.3 | Page 138

Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`

Miscellaneous Exercise 5 | Q 4.3 | Page 138

Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`

Miscellaneous Exercise 5 | Q 4.3 | Page 138

Evaluate: `int "dx"/("9x"^2 - 25)`

Miscellaneous Exercise 5 | Q 4.3 | Page 139

Evaluate: `int "e"^"x"/sqrt("e"^"2x" + 4"e"^"x" + 13)` dx

Miscellaneous Exercise 5 | Q 4.3 | Page 139

Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`

Miscellaneous Exercise 5 | Q 4.3 | Page 139

Evaluate: `int "dx"/(5 - 16"x"^2)`

Miscellaneous Exercise 5 | Q 4.3 | Page 139

Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`

Miscellaneous Exercise 5 | Q 4.3 | Page 139

Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx

Miscellaneous Exercise 5 | Q 4.4 | Page 139

Evaluate: `int (log "x")^2` dx

Miscellaneous Exercise 5 | Q 4.4 | Page 139

Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx

Miscellaneous Exercise 5 | Q 4.4 | Page 139

Evaluate: `int "x" * "e"^"2x"` dx

Miscellaneous Exercise 5 | Q 4.4 | Page 139

Evaluate: `int log ("x"^2 + "x")` dx

Miscellaneous Exercise 5 | Q 4.4 | Page 139

Evaluate: `int "e"^sqrt"x"` dx

Miscellaneous Exercise 5 | Q 4.4 | Page 139

Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx

Miscellaneous Exercise 5 | Q 4.4 | Page 139

Evaluate: `int sqrt("x"^2 - 8"x" + 7)` dx

Miscellaneous Exercise 5 | Q 4.5 | Page 139

Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx

Miscellaneous Exercise 5 | Q 4.5 | Page 139

Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx

Miscellaneous Exercise 5 | Q 4.5 | Page 139

Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx

Chapter 5: Integration

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6Miscellaneous Exercise 5
Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 5 - Integration

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 5 (Integration) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 5 Integration are Integration, Methods of Integration: Integration by Substitution, Methods of Integration: Integration by Parts, Methods of Integration: Integration Using Partial Fractions.

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