250+ TOP MCQs on Integration as an Inverse Process of Differentiation | Class 12 Maths

Mathematics Quiz for Engineering Entrance Exams on “Integration as an Inverse Process of Differentiation”.

1. Find the integral of (8x^3+1).
a) 2x⁴+x+C
b) 2x⁶-5x+C
c) 2x⁴-x+C
d) 2x⁴+x² C
Answer: a
Clarification: (int ,8x^{3+1} ,dx)
Using (int ,x^n ,dx=frac{x^{n+1}}{n+1}), we get
(int ,8x^{3+1} ,dx=int 8x^3 ,dx+int ,1 ,dx)
=(frac{8x^{3+1}}{3+1}+x)
=(frac{8x^4}{4}+x)
=2x⁴+x+C.

2. Find ∫ 7x²-x³+2x dx.
a) (frac{7x^3}{3}+frac{x^4}{5}-frac{2x^2}{2}+C)
b) (frac{7x^3}{3}+frac{x^4}{4}+frac{2x^2}{2}+C)
c) (frac{7x^5}{9}-frac{x^4}{4}+frac{2x^2}{2}+C)
d) (frac{7x^3}{3}-frac{x^4}{4}+x^2+C)
Answer: d
Clarification: To find (int 7x^2-x^3+2x dx)
(int 7x^2-x^3+2x dx=int 7x^2 dx-int x^3 dx+2int x dx)
Using (int x^n dx=frac{x^{n+1}}{n+1}), we get
(int 7x^2-x^3+2x dx=frac{7x^{2+1}}{2+1}-frac{x^{3+1}}{3+1}+2(frac{x^{1+1}}{1+1}))
∴(int 7x^2-x^3+2x dx=frac{7x^3}{3}-frac{x^4}{4}+x^2+C)

3. Find the integral of 2 sin⁡2x+3.
a) sin⁡2x+3x+C
b) -cos⁡2x-3x³+C
c) -cos⁡2x+3x+C
d) cos⁡2x-3x+12+C
Answer: c
Clarification: To find ∫ 2 sin⁡2x+3 dx
(int ,2 ,sin⁡2x+3 ,dx=int ,2 ,sin⁡2x ,dx + int ,3 ,dx)
(int ,2 ,sin⁡2x+3 ,dx=2int ,sin⁡2x ,dx+3int ,dx)
(int ,2 ,sin⁡2x+3 ,dx=frac{-2 cos⁡2x}{2}+3x)
∴∫2 sin⁡2x+3 dx=-cos⁡2x+3x+C

4. Find the integral of (int 3e^x+frac{2}{x}+x^3 dx).
a) (3e^3x+frac{2}{x}-frac{x^4}{4}+c)
b) (3e^x+2 ,log⁡x+frac{x^4}{4}+c)
c) (e^x+2 ,log⁡x+frac{x^4}{4}+c)
d) (3e^x-frac{2}{x^2}+frac{x^4}{4}+c)
Answer: b
Clarification: To find (int ,3e^x+frac{2}{x}+x^3 ,dx)
(int ,3e^x+frac{2}{x}+x^3 dx=3int ,e^x ,dx+2int frac{1}{x} ,dx+int x^3 ,dx)
(int ,e^x ,dx=e^x)
(int frac{1}{x} dx=log⁡x)
∴(int 3e^x+frac{2}{x}+x^3 ,dx=3e^x+2 ,log⁡x+frac{x^4}{4}+c)

5. Find the integral of (frac{4x^4-3x^2}{x^3}).
a) 7x²-3 log⁡x³+C
b) 2x²-3 log⁡x+C
c) x²-log⁡x+C
d) 2x²+3 log⁡x+C
Answer: b
Clarification: To find (int frac{4x^4-3x^2}{x^3} dx)
(int frac{4x^4-3x^2}{x^3} ,dx=int frac{4x^4}{x^3} – frac{3x^2}{x^3} ,dx)
(int frac{4x^4-3x^2}{x^3} ,dx=int 4x dx-int frac{3}{x} dx)
(int frac{4x^4-3x^2}{x^3} ,dx=frac{4x^2}{2}-3 log⁡x)
∴ (int frac{4x^4-3x^2}{x^3} ,dx=2x^2-3 ,log⁡x+C).

6. Find (int ,3 ,cos⁡x+frac{1}{x} dx).
a) (3 ,sin⁡x-frac{1}{x}+C)
b) (2 ,sin⁡x+frac{1}{x^3}+C)
c) (3 ,sin⁡3x+frac{1}{x}+C)
d) (sin⁡x-frac{1}{x^2}+C)
Answer: a
Clarification: To find (int ,3 ,cos⁡x+frac{1}{x^2} dx)
(int ,3 ,cos⁡x+frac{1}{x^2} dx=3 int cos⁡x ,dx+int frac{1}{x^2} ,dx)
(int ,3 ,cos⁡x+frac{1}{x^2} dx=3 ,sin⁡x+int x^{-2} ,dx)
(int ,3 ,cos⁡x+frac{1}{x^2} dx=3 ,sin⁡x+frac{x^{-2+1}}{-2+1})
(int ,3 ,cos⁡x+frac{1}{x^2} dx=3 ,sin⁡x-frac{1}{x}+C)

7. Find (int (2+x)xsqrt{x} dx).
a) (frac{4x^{5/2}}{5}+frac{2x^{7/2}}{9}+C)
b) (frac{4x^{5/2}}{5}-frac{2x^{7/2}}{7}+C)
c) (frac{4x^{5/2}}{6}+frac{2x^{7/2}}{7}+C)
d) –(frac{4x^{5/2}}{5}+frac{2x^{7/2}}{7}+C)
Answer: c
Clarification: To find (int (2+x)xsqrt{x} dx)
(int ,(2+x)xsqrt{x} ,dx=int ,2xsqrt{x}+x^{5/2} ,dx)
(int ,(2+x)xsqrt{x} ,dx=int ,2x^{3/2} dx + int x^{5/2} dx)
(int ,(2+x)xsqrt{x} ,dx=frac{2x^{3/2+1}}{3/2+1}+frac{x^{5/2+1}}{5/2+1})
(int ,(2+x)xsqrt{x} ,dx=frac{4x^{5/2}}{5}+frac{2x^{7/2}}{7}+C)

8. Find (int ,7x^8-4e^{2x}-frac{2}{x^2} ,dx).
a) (frac{7x^4}{4}-2e^{2x}+frac{2}{x}+C)
b) (frac{7x^4}{4}+2e^{2x}+frac{2}{x}+C)
c) (frac{7x^4}{4}-2e^{2x} frac{2}{x^2}+C)
d) (frac{7x^4}{8}+2e^{2x}-frac{4}{x}+C)
Answer: a
Clarification: To find:(int 7x^8-4e^{2x}-frac{2}{x^2} dx)
(int ,7x^8-4e^{2x}-frac{2}{x^2} ,dx=int 7x^9 dx-4int e^{2x} dx-2int frac{1}{x}^2 dx)
(int ,7x^8-4e^{2x}-frac{2}{x^2} ,dx=frac{7x^{9+1}}{9+1}-frac{4e^{2x}}{2}-frac{2x^{-2+1}}{-2+1})
∴(int ,7x^8-4e^{2x}-frac{2}{x^2} dx=frac{7x^{10}}{10}-2e^{2x}+frac{2}{x}+C)

9. Find the integral (int sin⁡2x+e^3x-cos⁡3x dx).
a) –(frac{sin⁡2x}{2}+frac{e^{3x}}{3}-frac{sin⁡3x}{3}+C)
b) –(frac{cos⁡2x}{2}+frac{e^{3x}}{3}-frac{sin⁡3x}{3}+C)
c) (frac{cos⁡2x}{2}+frac{e^{3x}}{3}-frac{cos⁡3x}{3}+C)
d) –(frac{cos⁡2x}{2}-frac{e^{3x}}{3}+frac{cos⁡3x}{3}+C)
Answer: b
Clarification: To find (int ,sin⁡2x+e^{3x}-cos⁡3x ,dx)
(int sin⁡2x+e^{3x}-cos⁡3x ,dx=int ,sin⁡2x ,dx+int ,e^{3x} ,dx-int ,cos⁡3x ,dx)
(int sin⁡2x+e^{3x}-cos⁡3x ,dx=-frac{cos⁡2x}{2}+frac{e^{3x}}{3}-frac{sin⁡3x}{3}+C)

10. Find the integral of (ax²+b)².
a) (frac{a^2 ,x^5}{5}+b^2 ,x+frac{2abx^3}{3}+C)
b) –(frac{a^2 ,x^5}{5}-b^2 ,x+frac{2abx^3}{3}+C)
c) (frac{b^2 ,x^5}{5}+b^2 x+frac{27x^3}{3}+C)
d) (frac{a^2 ,x^5}{5}+x+frac{2abx^3}{5}+C)
Answer: a
Clarification: To find (ax²+b)²
(int (ax^2+b)^2 dx=int (a^2 ,x^4+b^2+2ax^2 ,b) dx)
(int (ax^2+b)^2 dx=int ,a^2 ,x^4 ,dx+int ,b^2 ,dx+2int ,ax^2 ,b ,dx)
(int (ax^2+b)^2 dx=a^2 ,int ,x^4 ,dx+b^2 int ,dx+2abint ,x^2 ,dx)
(int (ax^2+b)^2 dx=a^2 (frac{x^5}{5})+b^2 x+2ab(frac{x^3}{3}))
(int (ax^2+b)^2 dx=frac{a^2 ,x^5}{5}+b^2 x+frac{2abx^3}{3}+C)

Mathematics Quiz for Engineering Entrance Exams,