This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Improper Integrals – 1”.
1. Integration of function is same as the ___________
a) Joining many small entities to create a large entity
b) Indefinitely small difference of a function
c) Multiplication of two function with very small change in value
d) Point where function neither have maximum value nor minimum value
View Answer
Explanation: Integration of function is same as the Joining many small entities to create a large entity.
2. Integration of (Sin(x) + Cos(x))ex is______________
a) ex Cos(x)
b) ex Sin(x)
c) ex Tan(x)
d) ex (Sin(x)+Cos(x))
View Answer
Explanation: Let f(x) = ex Sin(x)
∫ ex Sin(x)dx = ex Sin(x) – ∫ ex Cos(x)dx
∫ ex Sin(x)dx + ∫ ex Cos(x)dx = ∫ ex [Cos(x)+Sin(x)]dx = ex Sin(x).
3. Integration of (Sin(x) – Cos(x))ex is ___________
a) -ex Cos(x)
b) ex Cos(x)
c) -ex Sin(x)
d) ex Sin(x)
View Answer
Explanation: Add constant automatically
Let f(x) = ex Sin(x)
∫ ex Sin(x)dx = -ex Cos(x) + ∫ ex Cos(x)dx
∫ ex Sin(x)d-∫ ex Cos(x)dx = ∫ ex [Sin(x)-Cos(x)]dx = -ex Cos(x).
4. Value of ∫ Cos2 (x) Sin2 (x)dx.
a) \(\frac{1}{8} [x-\frac{Cos(2x)}{2}]\)
b) \(\frac{1}{4} [x-\frac{Cos(2x)}{2}]\)
c) \(\frac{1}{8} [x-\frac{Sin(2x)}{2}]\)
d) \(\frac{1}{4} [x-\frac{Sin(2x)}{2}]\)
View Answer
Explanation: Add constant automatically
Given,f(x)=\(\int Cos^2 (x) Sin^2 (x)dx=\frac{1}{4} \int Sin^2 (2x) dx=\frac{1}{4} \int \frac{[1-Cos(2x)]}{2} dx=\frac{1}{8} [x-\frac{Sin(2x)}{2}]\)
5. If differentiation of any function is zero at any point and constant at other points then it means?
a) Function is parallel to x-axis at that point
b) Function is parallel to y-axis at that point
c) Function is constant
d) Function is discontinuous at that point
View Answer
Explanation: Since slope of a function is given by dy⁄dx at that point. Hence, when dy⁄dx = 0 means slope of a function is zero i.e, parallel to x axis.
Function is not a constant function since it has finite value at other points.
6. If differentiation of any function is infinite at any point and constant at other points then it means ___________
a) Function is parallel to x-axis at that point
b) Function is parallel to y-axis at that point
c) Function is constant
d) Function is discontinuous at that point
View Answer
Explanation: Since slope of a function is given by dy⁄dx at that point.Hence,when dy⁄dx = ∞ means slope of a function is 90 degree i.e,parallel to y axis.
7. Integration of function y = f(x) from limit x1 < x < x2 , y1 < y < y2, gives ___________
a) Area of f(x) within x1 < x < x2
b) Volume of f(x) within x1 < x < x2
c) Slope of f(x) within x1 < x < x2
d) Maximum value of f(x) within x1 < x < x2
View Answer
Explanation: Integration of function y=f(x) from limit x1 < x < x2 , y1 < y < y2, gives area of f(x) within x1 < x < x2.
8. Find the value of ∫ ln(x)⁄x dx.
a) 3a2
b) a2
c) a
d) 1
View Answer
Explanation: Add constant automatically
Given, f(x)=\(\int \frac{ln(x)}{x} dx\)
Let, z=ln(x)=>dz=\(\frac{dx}{x}=>f(x)=\int zdz=z^2/2=\frac{ln^2(x)}{2}\)
9. Find the value of ∫t⁄(t+3)(t+2) dt, is?
a) 2 ln(t+3)-3 ln(t+2)
b) 2 ln(t+3)+3 ln(t+2)
c) 3 ln(t+3)-2 ln(t+2)
d) 3 ln(t+3)+2ln(t+2)
View Answer
Explanation: Add constant automatically
Given, et = x => dx = et dt,
Given, f(x)=\(\int \frac{ln(x)}{x} dx\)
Let, z=ln(x)=>dz=\(\frac{dx}{x}=>f(x)=\int zdz=\frac{z^2}{2}=\frac{ln^2(x)}{2}\)
10. Find the value of ∫ cot3(x) cosec4 (x).
a) –\([\frac{cot^4(x)}{4}+\frac{cosec^6(x)}{6}]\)
b) –\([\frac{cosec^4(x)}{4}+\frac{cosec^6(x)}{6}]\)
c) –\([\frac{cot^4(x)}{4}+\frac{cot^6(x)}{6}]\)
d) –\([\frac{cosec^4(x)}{4}+\frac{cot^6(x)}{6}]\)
View Answer
Explanation: Add constant automatically
Given, \(\int cot^3(x)cosec^4 (x)dx=-\int cot^3(x)cosec^2 (x)dcot(x)\)
=-\(\int t^3 (1+t^2)dt=-[\frac{t^4}{4}+\frac{t^6}{6}]=-[\frac{cot^4(x)}{4}+\frac{cot^6(x)}{6}]\)
11. Find the value of \(\int \frac{sec^4(x)}{\sqrt{tan(x)}} dx\).
a) \(\frac{2}{5}\sqrt{tan(x)}[5+sec^2(x)]\)
b) \(\frac{2}{5}\sqrt{sec(x)}[5+tan^2(x)]\)
c) \(\frac{2}{5}\sqrt{tan(x)}[6+tan^2(x)]\)
d) \(\frac{2}{5}\sqrt{tan(x)}[5+tan^2(x)]\)
View Answer
Explanation: Add constant automatically
Given, \(\int \frac{sec^4(x)}{\sqrt{tan(x)}} dx\)
=\(\int \frac{sec^2(x) sec^2(x)}{\sqrt{tan(x)}} dx\)
=\(\int \frac{1+t^2}{\sqrt{t}} dt\)
=\(\int [\frac{1}{\sqrt{t}}+t^{3/2}]dt\)
=\(2\sqrt{t}+\frac{2}{5} t^{5/2}\)
=\(\frac{2}{5}\sqrt{tan(x)}[5+tan^2(x)]\)
12. Find the value of \(\int \frac{1}{4x^2+4x+5} dx\).
a) 1⁄8 sin(-1)(x + 1⁄2)
b)1⁄4 tan(-1)(x + 1⁄2)
c) 1⁄8 sec(-1)(x + 1⁄2)
d) 1⁄4 cos(-1)(x + 1⁄2)
View Answer
Explanation: Add constant automatically
Given, \(\int \frac{1}{4x^2+4x+5} dx\)
=\(\int \frac{1}{4 (x^2+x+\frac{5}{4}+\frac{1}{4}+\frac{1}{4})} dx =\int \frac{1}{4[(x+\frac{1}{2})^2+1^2])}dx=\frac{1}{4} tan^{-1}(x+\frac{1}{2})\)
13. Find the value of \(\int \sqrt{4x^2+4x+5} dx\).
a) \(2\left [\frac{1}{2} (x+\frac{1}{2}) \sqrt{{(x+\frac{1}{2})^2+1)}}\right ]+ln\left [(x+\frac{1}{2})+\sqrt{(x+\frac{1}{2})^2+1} \right ]\)
b) \(2\left [\frac{1}{2} \sqrt{(x+\frac{1}{2})^2+1)}\right ]+\frac{1}{2} ln\left [(x+\frac{1}{2})+\sqrt{(x+\frac{1}{2})^2+1} \right ]\)
c) \(2\left [\frac{1}{2} (x+\frac{1}{2}) \sqrt{(x+\frac{1}{2})^2+1)}\right ]+\frac{1}{2} ln\left [(x+\frac{1}{2})+\sqrt{(x+\frac{1}{2})^2+1} \right ]\)
d) \(2\left [(x+\frac{1}{2}) \sqrt{{(x+\frac{1}{2})^2+1)}}\right ]+\frac{1}{2} ln\left [(x+\frac{1}{2})+\sqrt{(x+\frac{1}{2})^2+1} \right ]\)
View Answer
Explanation: Add constant automatically
Given, \(\int \sqrt{4x^2+4x+5} dx=\int 2\sqrt{(x+\frac{1}{2})^2+1^2} dx\)
=\(\int 2\sqrt{t^2+1^2} dt=2\left [\frac{1}{2} t\sqrt{t^2+1}\right ]+\frac{1}{2} ln[t+\sqrt{t^2+1}]\)
=\(2\left [\frac{1}{2} (x+\frac{1}{2}) \sqrt{(x+\frac{1}{2})^2+1)} \right ]+\frac{1}{2} ln\left [(x+\frac{1}{2})+\sqrt{(x+1/2)^2+1}\right ]\)
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