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## Chapter 5: Integration

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

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

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

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

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

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

If f '(x) = x^{2} + 5 and f(0) = - 1, then find the value of f(x).

If f '(x) = 4x^{3} - 3x^{2} + 2x + k, f(0) = 1 and f(1) = 4, find f(x).

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

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

∫ (x + 1)(x + 2)^{7} (x + 3)dx

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

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

**Evaluate the following.**

∫ x log x dx

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

**Evaluate the following.**

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

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

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

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

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

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

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

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

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

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

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

**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

**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

**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"`

**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

**Choose the correct alternative from the following.**

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

log x – log (1 – x) + c

log (1 - x

^{2}) + c- log x + log(1 - x) + c

log (x - x

^{2}) + c

**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

**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

**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"`

**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

**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

**Fill in the Blank.**

`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x^{5} + ______ x^{3} + 5x + c

**Fill in the Blank.**

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

**Fill in the Blank.**

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

**Fill in the Blank.**

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

**Fill in the Blank.**

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

**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

**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

**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

**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

**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

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

**Evaluate **`int (5"x" + 1)^(4/9)` dx

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

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

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

**Evaluate: **∫ |x| dx if x < 0

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

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

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

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

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

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

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

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

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

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

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

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

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

**Evaluate: **`int (log "x")^2` dx

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

**Evaluate: **`int "x" * "e"^"2x"` dx

**Evaluate: **`int log ("x"^2 + "x")` dx

**Evaluate: **`int "e"^sqrt"x"` dx

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

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

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

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

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

## Chapter 5: Integration

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

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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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