Class 12 Maths Chapter 7 Integrals MCQ

This set of Class 12 Maths Chapter 7 Multiple Choice Questions & Answers (MCQs) focuses on “Integrals”. These MCQs are created based on the latest CBSE syllabus and the NCERT curriculum, offering valuable assistance for exam preparation.

1. Find the integral of \(8x^3+1\).
a) 2x⁴+x+C
b) 2x⁶-5x+C
c) 2x⁴-x+C
d) 2x⁴+x² C
View Answer

Answer: a
Explanation: \(\int \,8x^{3+1} \,dx\)
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) \(\frac{7x^3}{3}-\frac{x^4}{4}+x^2+C\)
View Answer

Answer: d
Explanation: To find \(\int 7x^2-x^3+2x dx\)
\(\int 7x^2-x^3+2x dx=\int 7x^2 dx-\int x^3 dx+2\int x dx\)
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
View Answer

Answer: c
Explanation: To find ∫ 2 sin⁡2x+3 dx
\(\int \,2 \,sin⁡2x+3 \,dx=\int \,2 \,sin⁡2x \,dx + \int \,3 \,dx\)
\(\int \,2 \,sin⁡2x+3 \,dx=2\int \,sin⁡2x \,dx+3\int \,dx\)
\(\int \,2 \,sin⁡2x+3 \,dx=\frac{-2 cos⁡2x}{2}+3x\)
∴∫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) \(3e^x-\frac{2}{x^2}+\frac{x^4}{4}+c\)
View Answer

Answer: b
Explanation: To find \(\int \,3e^x+\frac{2}{x}+x^3 \,dx\)
\(\int \,3e^x+\frac{2}{x}+x^3 dx=3\int \,e^x \,dx+2\int \frac{1}{x} \,dx+\int x^3 \,dx\)
\(\int \,e^x \,dx=e^x\)
\(\int \frac{1}{x} dx=log⁡x\)
∴\(\int 3e^x+\frac{2}{x}+x^3 \,dx=3e^x+2 \,log⁡x+\frac{x^4}{4}+c\)

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
View Answer

Answer: b
Explanation: To find \(\int \frac{4x^4-3x^2}{x^3} dx\)
\(\int \frac{4x^4-3x^2}{x^3} \,dx=\int \frac{4x^4}{x^3} – \frac{3x^2}{x^3} \,dx\)
\(\int \frac{4x^4-3x^2}{x^3} \,dx=\int 4x dx-\int \frac{3}{x} dx\)
\(\int \frac{4x^4-3x^2}{x^3} \,dx=\frac{4x^2}{2}-3 log⁡x\)
∴ \(\int \frac{4x^4-3x^2}{x^3} \,dx=2x^2-3 \,log⁡x+C\).

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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) \(sin⁡x-\frac{1}{x^2}+C\)
View Answer

Answer: a
Explanation: 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 \(\int (2+x)x\sqrt{x} dx\).
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) –\(\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C\)
View Answer

Answer: c
Explanation: To find \(\int (2+x)x\sqrt{x} dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x\sqrt{x}+x^{5/2} \,dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x^{3/2} dx + \int x^{5/2} dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\frac{2x^{3/2+1}}{3/2+1}+\frac{x^{5/2+1}}{5/2+1}\)
\(\int \,(2+x)x\sqrt{x} \,dx=\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C\)

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) \(\frac{7x^4}{8}+2e^{2x}-\frac{4}{x}+C\)
View Answer

Answer: a
Explanation: To find:\(\int 7x^8-4e^{2x}-\frac{2}{x^2} dx\)
\(\int \,7x^8-4e^{2x}-\frac{2}{x^2} \,dx=\int 7x^9 dx-4\int e^{2x} dx-2\int \frac{1}{x}^2 dx\)
\(\int \,7x^8-4e^{2x}-\frac{2}{x^2} \,dx=\frac{7x^{9+1}}{9+1}-\frac{4e^{2x}}{2}-\frac{2x^{-2+1}}{-2+1}\)
∴\(\int \,7x^8-4e^{2x}-\frac{2}{x^2} dx=\frac{7x^{10}}{10}-2e^{2x}+\frac{2}{x}+C\)

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) –\(\frac{cos⁡2x}{2}-\frac{e^{3x}}{3}+\frac{cos⁡3x}{3}+C\)
View Answer

Answer: b
Explanation: 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) \(\frac{a^2 \,x^5}{5}+x+\frac{2abx^3}{5}+C\)
View Answer

Answer: a
Explanation: To find (ax²+b)²
\(\int (ax^2+b)^2 dx=\int (a^2 \,x^4+b^2+2ax^2 \,b) dx\)
\(\int (ax^2+b)^2 dx=\int \,a^2 \,x^4 \,dx+\int \,b^2 \,dx+2\int \,ax^2 \,b \,dx\)
\(\int (ax^2+b)^2 dx=a^2 \,\int \,x^4 \,dx+b^2 \int \,dx+2ab\int \,x^2 \,dx\)
\(\int (ax^2+b)^2 dx=a^2 (\frac{x^5}{5})+b^2 x+2ab(\frac{x^3}{3})\)
\(\int (ax^2+b)^2 dx=\frac{a^2 \,x^5}{5}+b^2 x+\frac{2abx^3}{3}+C\)

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