Questions and Solutions
Question 1:
Find the integral: \( \int \frac{x^3 + 3x + 4}{\sqrt{x}} \, dx \)
Question 2:
Find the integral: \( \int \frac{x^3 - x^2 + x - 1}{x - 1} \, dx \)
Question 3:
Find the integral: \( \int (1 - x) \sqrt{x} \, dx \)
Question 4:
Find the integral: \( \int \sqrt{x} (3x^2 + 2x + 3) \, dx \)
Question 5:
Find the integral: \( \int (2x - 3\cos x + e^x) \, dx \)
Question 6:
Find the integral: \( \int (2x^2 - 3\sin x + 5\sqrt{x}) \, dx \)
Question 7:
Find the integral: \( \int \sec x (\sec x + \tan x) \, dx \)
Question 8:
Find the integral: \( \int \frac{\sec^2 x}{\csc^2 x} \, dx \)
Question 9:
Find the integral: \( \int \frac{2 - 3\sin x}{\cos^2 x} \, dx \)
Question 10:
The anti-derivative of \( \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \) is: (a) \( \frac{1}{3}x^{1/3} + 2x^{1/2} + C \) (b) \( \frac{2}{3}x^{2/3} + \frac{1}{2}x^2 + C \) (c) \( \frac{2}{3}x^{3/2} + 2x^{1/2} + C \) (d) \( \frac{3}{2}x^{3/2} + \frac{1}{2}x^{1/2} + C \)
Question 11:
If \( \frac{d}{dx}f(x) = 4x^3 - \frac{3}{x^4} \) and \( f(2) = 0 \), then \( f(x) \) is: (a) \( x^4 + \frac{1}{x^3} - \frac{129}{8} \) (b) \( x^3 + \frac{1}{x^4} + \frac{129}{8} \) (c) \( x^4 + \frac{1}{x^3} + \frac{129}{8} \) (d) \( x^3 + \frac{1}{x^4} - \frac{129}{8} \)
Question 12:
Integrate \( \dfrac{2x}{1+x^2} \)
Question 13:
Integrate \( \dfrac{(\log x)^2}{x} \)
Question 14:
Integrate \( \dfrac{1}{x + x \log x} \)
Question 15:
Integrate \( \sin x \cdot \sin(\cos x) \)
Question 16:
Integrate \( \sin(ax+b)\cos(ax+b) \)
Question 17:
Integrate \( \sqrt{ax+b} \)
Question 18:
Integrate \( x\sqrt{x+2} \)
Question 19:
Integrate \( x\sqrt{1+2x^2} \)
Question 20:
Integrate \( (4x + 2)\sqrt{x^2 + x + 1} \)
Question 21:
Integrate \( \dfrac{1}{x - \sqrt{x}} \)
Question 22:
Integrate \( \dfrac{x}{\sqrt{x + 4}},\ x > 0 \)
Question 23:
Integrate \( (x^3 - 1)^{1/3}x^5 \)
Question 24:
Integrate \( \dfrac{x^2}{(2 + 3x^3)^3} \)
Question 25:
Integrate \( \dfrac{1}{x(\log x)^m},\ x > 0 \)
Question 26:
Integrate \( \dfrac{x}{9 - 4x^2} \)
Question 27:
Integrate \( e^{2x+3} \)
Question 28:
Integrate \( \dfrac{x}{e^{x^2}} \)
Question 29:
Integrate \( \dfrac{e^{\tan^{-1} x}}{1 + x^2} \)
Question 30:
Integrate \( \dfrac{e^{2x} - 1}{e^{2x} + 1} \)
Question 31:
Evaluate: \( \int \dfrac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}} dx \)
Question 32:
Evaluate: \( \int \tan^2(2x - 3) dx \)
Question 33:
Evaluate: \( \int \sec^2(7 - 4x) dx \)
Question 34:
Evaluate: \( \int \dfrac{\sin^{-1} x}{\sqrt{1 - x^2}} dx \)
Question 35:
Evaluate: \( \int \dfrac{2\cos x - 3\sin x}{6\cos x + 4\sin x} dx \)
Question 36:
Evaluate: \( \int \dfrac{1}{\cos^2 x (1 - \tan x)^2} dx \)
Question 37:
Evaluate: \( \int \dfrac{\cos \sqrt{x}}{\sqrt{x}} dx \)
Question 38:
Evaluate: \( \int \sqrt{\sin 2x} \cos 2x dx \)
Question 39:
Evaluate: \( \int \dfrac{\cos x}{\sqrt{1 + \sin x}} dx \)
Question 40:
Evaluate: \( \int \cot x \log \sin x dx \)
Question 41:
Evaluate: \( \int \dfrac{\sin x}{1 + \cos x} dx \)
Question 42:
Evaluate: \( \int \dfrac{\sin x}{(1 + \cos x)^2} dx \)
Question 43:
Evaluate: \( \int \dfrac{1}{1 - \tan x} dx \)
Question 44:
Evaluate: \( \int \dfrac{\sqrt{\tan x}}{\sin x \cos x} dx \)
Question 45:
Evaluate: \( \int \dfrac{(1 + \log x)^2}{x} dx \)
Question 46:
Evaluate: \( \int \dfrac{(x+1)(x + \log x)^2}{x} dx \)
Question 47:
Evaluate: \( \int \dfrac{x^3 \sin(\tan^{-1} x^4)}{1 + x^8} dx \)
Question 48:
Evaluate: \( \int \dfrac{10x^9 + 10^x \log_{e} 10}{x^{10} + 10^x} dx \)
Question 49:
Evaluate: \( \int \dfrac{dx}{\sin^2 x \cos^2 x} \)
Question 50:
Integrate: \( \sin^2(2x + 5) \)
Question 51:
Integrate: \( \sin 3x \cos 4x \)
Question 52:
Integrate: \( \cos 2x \cos 4x \cos 6x \)
Question 53:
Integrate: \( \sin^3(2x + 1) \)
Question 54:
Integrate: \( \sin^3 x \cos^3 x \)
Question 55:
Integrate: \( \sin x \sin 2x \sin 3x \)
Question 56:
Integrate: \( \sin 4x \sin 8x \)
Question 57:
Integrate: \( \frac{1 - \cos x}{1 + \cos x} \)
Question 58:
Integrate: \( \frac{\cos x}{1 + \cos x} \)
Question 59:
Integrate: \( \sin^4 x \)
Question 60:
Integrate: \( \frac{\sin^2 x}{1 + \cos x} \)
Question 61:
Integrate: \( \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha} \)
Question 62:
Integrate: \( \frac{\cos x - \sin x}{1 + \sin 2x} \)
Question 63:
Integrate: \( \tan^3 2x \sec 2x \)
Question 64:
Integrate: \( \tan^4 x \)
Question 65:
Integrate: \( \frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} \)
Question 66:
Integrate: \( \frac{\cos 2x + 2\sin^2 x}{\cos^2 x} \)
Question 67:
Integrate: \( \frac{1}{\sin x \cos^3 x} \)
Question 68:
Integrate: \( \frac{\cos 2x}{(\cos x + \sin x)^2} \)
Question 69:
Integrate: \( \sin^{-1}(\cos x) \)
Question 70:
Integrate: \( \frac{1}{\cos(x - a) \cos(x - b)} \)
Question 71:
Evaluate: \( \int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx \)
Question 72:
Evaluate: \( \int \frac{e^x(1+x)}{\cos^2(e^x x)} dx \)
Question 73:
Evaluate: \(\int \dfrac{1}{{{x}^{2}}-9}\)
Question 74:
Evaluate: \(\int \dfrac{3x-1}{\left( x-1 \right)\left( x-2 \right)\left( x-3 \right)}\)
Question 75:
Evaluate: \( \int \dfrac{x}{\left( x-1 \right)\left( x-2 \right)\left( x-3 \right)}\)
Question 76:
Evaluate: \(\int \dfrac{2x}{{{x}^{2}}+3x+2}\)
Question 77:
Evaluate: \(\int \dfrac{1-{{x}^{2}}}{x\left( 1-2x \right)}\)
Question 78:
Evaluate: \(\int \dfrac{x}{\left( {{x}^{2}}+1 \right)\left( x-1 \right)}\)
Question 79:
Evaluate: \(\int \dfrac{x}{{{\left( x-1 \right)}^{2}}\left( x+2 \right)}\)
Question 80:
Evaluate: \(\int \dfrac{3x+5}{{{x}^{3}}-{{x}^{2}}-x+1}\)
Question 81:
Evaluate: \(\int \dfrac{2x-3}{\left( {{x}^{2}}-1 \right)\left( 2x+3 \right)}\)
Question 82:
Evaluate: \(\int \dfrac{5x}{\left( x+1 \right)\left( {{x}^{2}}-4 \right)}\)
Question 83:
Evaluate: \(\int \dfrac{{{x}^{2}}+x+1}{{{x}^{2}}-1}\)
Question 84:
Evaluate: \(\int \dfrac{2}{\left( 1-x \right)\left( 1+{{x}^{2}} \right)}\)
Question 85:
Evaluate: \(\int \dfrac{3x-1}{{{\left( x+2 \right)}^{2}}}\)
Question 86:
Evaluate: \(\int \dfrac{1}{{{x}^{4}}-1}\)
Question 87:
Evaluate: \(\int \dfrac{1}{x\left( {{x}^{n}}+1 \right)}\) [Hint: Multiply numerator and denominator by \(x^{n-1}\) and put \(x^n = t\)]
Question 88:
Evaluate: \(\int \dfrac{\cos x}{\left( 1-\sin x \right)\left( 2-\sin x \right)}\) [Hint: Put \(\sin x = t\)]
Question 89:
Evaluate: \(\int \dfrac{\left( {{x}^{2}}+1 \right)\left( {{x}^{2}}+2 \right)}{\left( {{x}^{2}}+3 \right)\left( {{x}^{2}}+4 \right)}\)
Question 90:
Evaluate: \(\int \dfrac{2x}{\left( {{x}^{2}}+1 \right)\left( {{x}^{2}}+3 \right)}\)
Question 91:
Evaluate: \( \int \dfrac{1}{x\left( {{x}^{4}}-1 \right)}\)
Question 92:
Evaluate: \(\int \dfrac{1}{{{e}^{x}}-1}\) [Hint: Put \(e^x = t\)]
Question 93:
Evaluate: \(\int{\dfrac{xdx}{\left( x-1 \right)\left( x-2 \right)}}\)
Question 94:
Evaluate: \(\int{\dfrac{dx}{x\left( {{x}^{2}}+1 \right)}}\)
Question 95:
Integrate \(x\sin x\)
Question 96:
Integrate \(x\sin 3x\)
Question 97:
Integrate \({{x}^{2}}{{e}^{x}}\)
Question 98:
Integrate \(x\log x\)
Question 99:
Integrate \(x\log 2x\)
Question 100:
Integrate \({{x}^{2}}\log x\)
Question 101:
Integrate \(x{{\sin }^{-1}}x\)
Question 102:
Integrate \(x{{\tan }^{-1}}x\)
Question 103:
Integrate \(x{{\cos }^{-1}}x\)
Question 104:
Integrate \({{\left( {{\sin }^{-1}}x \right)}^{2}}\)
Question 105:
Integrate \(\dfrac{x{{\cos }^{-1}}x}{\sqrt{1-{{x}^{2}}}}\)
Question 106:
Integrate \(x{{\sec }^{2}}x\)
Question 107:
Integrate \({{\tan }^{-1}}x\)
Question 108:
Integrate \(x{{\left( \log x \right)}^{2}}dx\)
Question 109:
Integrate \(\left( {{x}^{2}}+1 \right)\log x\)
Question 110:
Integrate \({{e}^{x}}\left( \sin x+\cos x \right)\)
Question 111:
Integrate \(\dfrac{x{{e}^{x}}}{{{\left( 1+x \right)}^{2}}}\)
Question 112:
Integrate \({{e}^{x}}\left( \dfrac{1+\sin x}{1+\cos x} \right)\)
Question 113:
Integrate \({{e}^{x}}\left( \dfrac{1}{x}-\dfrac{1}{{{x}^{2}}} \right)\)
Question 114:
Integrate \(\dfrac{(x-3){{e}^{x}}}{{{(x-1)}^{3}}}\)
Question 115:
Integrate \({{e}^{2x}}\sin x\)
Question 116:
Integrate \({{\sin }^{-1}}\left( \dfrac{2x}{1+{{x}^{3}}} \right)\)
Question 117:
Integrate \(\sqrt{4-{{x}^{2}}}\)
Question 118:
Integrate \(\sqrt{1-4{{x}^{2}}}\)
Question 119:
Integrate \(\sqrt{{{x}^{2}}+4x+6}\)
Question 120:
Integrate \(\sqrt{{{x}^{2}}+4x+1}\)
Question 121:
Integrate \(\sqrt{1-4x-{{x}^{2}}}\)
Question 122:
Integrate \(\sqrt{{{x}^{2}}+4x-5}\)
Question 123:
Integrate \(\sqrt{1+3x-{{x}^{2}}}\)
Question 124:
Integrate \(\sqrt{{{x}^{2}}+3x}\)
Question 125:
Integrate \(\sqrt{1+\dfrac{{{x}^{2}}}{9}}\)