Related InterviewSolutions

• Let n ≥ 2 be a natural number and \(0 < \theta < \frac{\pi }{2}.{\rm{\;Then\;}}\smallint \left( {\frac{{{{\left( {{\rm{si}}{{\rm{n}}^n}\theta - {\rm{sin}}\theta } \right)}^{\frac{1}{n}}}{\rm{cos}}\theta }}{{{\rm{si}}{{\rm{n}}^{n + 1}}\theta }}} \right)d\theta\)is equal to:(Where C is a constant of integration)1. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}\theta }}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)2. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} + 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}{\rm{\theta }}}}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)3. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 + \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}{\rm{\theta }}}}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)4. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} + 1}}\theta }}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\)
• \(\smallint \frac{{{\rm{sin}}\frac{{5{\rm{x}}}}{2}}}{{{\rm{sin}}\frac{{\rm{x}}}{2}}}{\rm{dx}}\)is equal to(where c is a constant of integration.)1. 2x + sin x + 2 sin 2x + C2. x + 2 sin x + 2 sin 2x + C3. x + 2 sin x + sin 2x + C4. 2x + sin x – sin 2x + C
• If\(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^3}{\rm{dx}} = 0{\rm{\;}}\) and \(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^2}{\rm{dx}} = \frac{2}{3},\)then what are the values of a and b respectively?1. -1, 12. 1, 13. 0, 04. 2, -25. None of these
• The integral\(\;\mathop \smallint \nolimits_{\pi /6}^{\pi /3} {\rm{se}}{{\rm{c}}^{2/3}}x{\rm{cose}}{{\rm{c}}^{4/3}}x\;dx{\rm{\;is}}\)equal to:1. 35⁄6 - 32⁄32. 34⁄3 - 31⁄33. 37⁄6 - 35⁄64. 35⁄3 - 31⁄3
• Find a + b, if\(\smallint \left( {\sin 2x - 4{e^{3x}}} \right)\;dx = - \frac{1}{a}\cos 2x - \frac{b}{3}\;{e^{3x}} + C\)where C is the integration constant ?1. 62. 43. 24. None of these
• Find a + 3b if\(\smallint \left( {{x^2} - 1} \right)\;dx = \frac{{{x^3}}}{a} + bx + C\)where C is integration constant ?1. 22. 03. 14. None of these
• A value ofα such that \(\mathop \smallint \nolimits_\alpha ^{\alpha + 1} \frac{{{\rm{d}}x}}{{\left( {x + \alpha } \right)\left( {x + \alpha + 1} \right)}} = {\rm{lo}}{{\rm{g}}_{\rm{e}}}\left( {\frac{9}{8}} \right)\)is:1. -22. \(\frac{1}{2}\)3. \(- \frac{1}{2}\)4. 2
• The integral \(\int \left( \frac{2{{\text{x}}^{3}}-1}{{{\text{x}}^{4}}+\text{x}} \right)\text{dx}\)is equal to:(Here C is a constant of integration)1. \(\frac{1}{2}\text{lo}{{\text{g}}_{\text{e}}}\frac{\left| {{\text{x}}^{3}}+1 \right|}{{{\text{x}}^{2}}}+\text{C}\)2. \(\frac{1}{2}\text{lo}{{\text{g}}_{\text{e}}}\frac{{{\left( {{\text{x}}^{3}}+1 \right)}^{2}}}{\left| {{\text{x}}^{3}} \right|}+\text{C}\)3. \(\text{lo}{{\text{g}}_{\text{e}}}\left| \frac{{{\text{x}}^{3}}+1}{\text{x}} \right|+\text{C}\)4. \(\text{lo}{{\text{g}}_{\text{e}}}\frac{\left| {{\text{x}}^{3}}+1 \right|}{{{\text{x}}^{2}}}+\text{C}\)
• if\(\displaystyle\int\dfrac{\sin x}{\cos (x-a)}dx=Ax+B\log |sec(x-a)|+c\)where A, B and c are real constants then:1. A = sin a, B∈ℝ, c = cos a2. A = cos a, B = sin a, c∈ℝ3. A∈ℝ, B = cos a, c = sin a4. A = sin a, B = cos a, c∈ℝ
• Evaluation of\(\displaystyle\int{\dfrac{1-\tan x}{1+\tan x}}\ dx\)is:1. log (sin x - cos x) + c2. log (sin x - cot x) + c3. log (cos x + sin x) + c4. log (cos x - cot x) + c