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

## Chapter 5: Integration

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

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 Integration Exercise 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 Integration Exercise 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 Integration Exercise 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 [Page 129]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 5 Integration Exercise 5.4 [Page 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 Integration Exercise 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 Integration Exercise 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 Integration Miscellaneous 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) + 4e^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

## 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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