Mathematics Questions and Answers – Integration as an Inverse Process of Differentiation

«
»

This set of Mathematics Quiz for Engineering Entrance Exams focuses 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
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.
advertisement

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
advertisement
advertisement

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 \(\int \frac{4x^4-3x^2}{x^3} dx\)
\(\int \frac{4x^4-3x^2}{x^3} \,dx=\int \frac{4x^4}{x^3} – \frac{3x^2}{x^3} \,dx\)
\(\int \frac{4x^4-3x^2}{x^3} \,dx=\int 4x dx-\int \frac{3}{x} dx\)
\(\int \frac{4x^4-3x^2}{x^3} \,dx=\frac{4x^2}{2}-3 log⁡x\)
∴ \(\int \frac{4x^4-3x^2}{x^3} \,dx=2x^2-3 \,log⁡x+C\).
advertisement

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 \(\int (2+x)x\sqrt{x} dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x\sqrt{x}+x^{5/2} \,dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\int \,2x^{3/2} dx + \int x^{5/2} dx\)
\(\int \,(2+x)x\sqrt{x} \,dx=\frac{2x^{3/2+1}}{3/2+1}+\frac{x^{5/2+1}}{5/2+1}\)
\(\int \,(2+x)x\sqrt{x} \,dx=\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C\)
advertisement

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 \(\int \,sin⁡2x+e^{3x}-cos⁡3x \,dx\)
\(\int sin⁡2x+e^{3x}-cos⁡3x \,dx=\int \,sin⁡2x \,dx+\int \,e^{3x} \,dx-\int \,cos⁡3x \,dx\)
\(\int sin⁡2x+e^{3x}-cos⁡3x \,dx=-\frac{cos⁡2x}{2}+\frac{e^{3x}}{3}-\frac{sin⁡3x}{3}+C\)
advertisement

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
\(\int (ax^2+b)^2 dx=\int (a^2 \,x^4+b^2+2ax^2 \,b) dx\)
\(\int (ax^2+b)^2 dx=\int \,a^2 \,x^4 \,dx+\int \,b^2 \,dx+2\int \,ax^2 \,b \,dx\)
\(\int (ax^2+b)^2 dx=a^2 \,\int \,x^4 \,dx+b^2 \int \,dx+2ab\int \,x^2 \,dx\)
\(\int (ax^2+b)^2 dx=a^2 (\frac{x^5}{5})+b^2 x+2ab(\frac{x^3}{3})\)
\(\int (ax^2+b)^2 dx=\frac{a^2 \,x^5}{5}+b^2 x+\frac{2abx^3}{3}+C\)

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

To practice Mathematics Quiz for Engineering Entrance Exams, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

advertisement
advertisement
Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter