GIKI Entry Test Mathematics Integration — Set 3

GIKI Entry Test Mathematics Integration — Set 3

1. Question 1

   Q1. ∫(x + 1) / (x² + 2x + 1) dx = ?

   • A) (1/2)ln|x² + 2x + 1| + C
   • B) ln|x + 1| + C
   • C) (1/2)ln|x + 1| + C
   • D) x + ln|x² + 2x + 1| + C

   Answer: ln|x + 1| + C

   Explanation: Since x² + 2x + 1 = (x + 1)², the integral simplifies to ∫(x + 1) / (x + 1)² dx = ∫1 / (x + 1) dx.

2. Question 2

   Q2. ∫e^(-x) dx = ?

   • A) -e^(-x) + C
   • B) e^(-x) + C
   • C) -e^(x) + C
   • D) e^(x) + C

   Answer: -e^(-x) + C

   Explanation: The integral of e^(-x) is -e^(-x) since the derivative of -x is -1.

3. Question 3

   Q3. ∫(1 / √(4 - x²)) dx = ?

   • A) sin^-1(x/2) + C
   • B) cos^-1(x/2) + C
   • C) tan^-1(x/2) + C
   • D) sec^-1(x/2) + C

   Answer: sin^-1(x/2) + C

   Explanation: The given integral is the standard form of the derivative of sin^-1(x/2).

4. Question 4

   Q4. ∫(x + 2) dx from 0 to 1 = ?

   • A) 5/2
   • B) 3/2
   • C) 2
   • D) 5

   Answer: 5/2

   Explanation: Evaluating the integral gives [(x²/2) + 2x] from 0 to 1 = (1/2 + 2) - 0 = 5/2.

5. Question 5

   Q5. ∫(e^x + e^-x) dx = ?

   • A) e^x - e^-x + C
   • B) e^x + e^-x + C
   • C) e^x - e^x + C
   • D) e^-x + e^x + C

   Answer: e^x - e^-x + C

   Explanation: The integral of e^x is e^x and the integral of e^-x is -e^-x.

6. Question 6

   Q6. ∫x³ dx = ?

   • A) (x⁴)/4 + C
   • B) (x⁴)/3 + C
   • C) (x³)/3 + C
   • D) (x²)/2 + C

   Answer: (x⁴)/4 + C

   Explanation: Using the power rule of integration, ∫x^n dx = x^(n+1)/(n+1) + C.

7. Question 7

   Q7. ∫x² sinx dx = ?

   • A) -x² cosx + 2x sinx + 2 cosx + C
   • B) -x² cosx + 2x sinx - 2 cosx + C
   • C) x² cosx + 2x sinx + 2 cosx + C
   • D) x² cosx - 2x sinx + 2 cosx + C

   Answer: -x² cosx + 2x sinx + 2 cosx + C

   Explanation: Using integration by parts twice, we arrive at the given solution.

8. Question 8

   Q8. ∫x / (x + 1) dx = ? on 1 / 2 to 1

   • A) 1 / 2 - ln(3 / 2)
   • B) 1 / 2 + ln(3 / 2)
   • C) 1 - ln(2)
   • D) 1 + ln(2)

   Answer: 1 / 2 - ln(3 / 2)

   Explanation: ∫x / (x + 1) dx = ∫1 - 1/(x+1) dx = x - ln|x+1|; evaluate from 1/2 to 1.

9. Question 9

   Q9. ∫(1 + x) / (√x) dx = ?

   • A) (2/3)x^(3/2) + 2√x + C
   • B) 2√x + (2/3)x^(3/2) + C
   • C) (2/3)x^(3/2) - 2√x + C
   • D) 2√x - (2/3)x^(3/2) + C

   Answer: 2√x + (2/3)x^(3/2) + C

   Explanation: Rewrite as ∫(x^(-1/2) + x^(1/2)) dx = 2x^(1/2) + (2/3)x^(3/2) + C.

10. Question 10

    Q10. ∫(sinx + cosx)² dx = ?

    • A) x + sin²x + C
    • B) x - sin²x + C
    • C) x + sin(2x) + C
    • D) x - sin(2x) + C

    Answer: x + sin(2x) + C

    Explanation: Expand (sinx + cosx)² = 1 + sin(2x), then integrate term by term.

11. Question 11

    Q11. ∫(x + 1) / (x² + 2x + 5) dx = ?

    • A) 1 / 2 ln|x² + 2x + 5| + C
    • B) 1 / 2 ln|x² + 2x + 5| + (1/2)tan^(-1)((x+1)/2) + C
    • C) (1/2)tan^(-1)((x+1)/2) + C
    • D) ln|x² + 2x + 5| + C

    Answer: 1 / 2 ln|x² + 2x + 5| + (1/2)tan^(-1)((x+1)/2) + C

    Explanation: Rewrite the numerator to match the derivative of the denominator and use arctan form.

12. Question 12

    Q12. ∫e^x (sinx + cosx) dx = ?

    • A) e^x sinx + C
    • B) e^x cosx + C
    • C) e^x (sinx - cosx) + C
    • D) e^x (cosx - sinx) + C

    Answer: e^x sinx + C

    Explanation: Using integration by parts, we simplify to e^x sinx + C.

13. Question 13

    Q13. ∫(1 + tanx) / (1 - tanx) dx = ?

    • A) -ln|cos(x - π/4)| + C
    • B) ln|cos(x + π/4)| + C
    • C) ln|sin(x + π/4)| + C
    • D) -ln|sin(x - π/4)| + C

    Answer: -ln|cos(x - π/4)| + C

    Explanation: Using the identity tan(A-B), simplify the expression to tan(x + π/4).

14. Question 14

    Q14. ∫(x + 3) / √(x² + 4x + 13) dx = ?

    • A) √(x² + 4x + 13) + ln|x + 2 + √(x² + 4x + 13)| + C
    • B) √(x² + 4x + 13) + C
    • C) 1/2 √(x² + 4x + 13) + ln|x + 2 + √(x² + 4x + 13)| + C
    • D) 1/2 √(x² + 4x + 13) + C

    Answer: √(x² + 4x + 13) + ln|x + 2 + √(x² + 4x + 13)| + C

    Explanation: Split the integral into two parts, one simplifies to √(x²+4x+13) and the other to ln|x+2+√(x²+4x+13)|.

15. Question 15

    Q15. ∫(sin(√x)) / √x dx = ?

    • A) -2 cos(√x) + C
    • B) 2 cos(√x) + C
    • C) -2 sin(√x) + C
    • D) 2 sin(√x) + C

    Answer: -2 cos(√x) + C

    Explanation: Substitute u = √x, du = 1/(2√x) dx, then 2∫sin(u) du = -2cos(u) + C.

16. Question 16

    Q16. ∫(2x + 1) dx from 0 to 1 = ?

    • A) 1
    • B) 2
    • C) 3 / 2
    • D) 2 / 3

    Answer: 3 / 2

    Explanation: Using the power rule of integration, ∫(2x + 1) dx = x² + x. Evaluate from 0 to 1: (1² + 1) - (0² + 0) = 2.

17. Question 17

    Q17. ∫sin(x) dx from 0 to π = ?

    • A) 0
    • B) 1
    • C) 2
    • D) -1

    Answer: 2

    Explanation: The antiderivative of sin(x) is -cos(x). Evaluate from 0 to π: (-cos(π)) - (-cos(0)) = -(-1) - (-1) = 2.

18. Question 18

    Q18. ∫(1 / x) dx from 1 to 2 = ?

    • A) log(2)
    • B) log(1 / 2)
    • C) 1 / 2
    • D) 1

    Answer: log(2)

    Explanation: The antiderivative of 1 / x is log|x|. Evaluate from 1 to 2: log(2) - log(1) = log(2).

19. Question 19

    Q19. ∫(x² + 1) dx from 0 to 1 = ?

    • A) 4 / 3
    • B) 3 / 4
    • C) 1 / 3
    • D) 2 / 3

    Answer: 4 / 3

    Explanation: Using the power rule of integration, ∫(x² + 1) dx = (1 / 3)x³ + x. Evaluate from 0 to 1: ((1 / 3)(1)³ + 1) - ((1 / 3)(0)³ + 0) = 4 / 3.

20. Question 20

    Q20. ∫e^x dx from 0 to 1 = ?

    • A) e - 1
    • B) e + 1
    • C) 1 - e
    • D) e

    Answer: e - 1

    Explanation: The antiderivative of e^x is e^x. Evaluate from 0 to 1: e^1 - e^0 = e - 1.