CBSE Class 12 Mathematics Integrals VBQs Set 03

Short Answer Questions :

Question. Evaluate: \(\int x \sin^{-1} x dx\) 
Answer: \(\frac{x^2}{2} \sin^{-1} x - \frac{1}{4} \sin^{-1} x + \frac{x}{4} \sqrt{1 - x^2} + C\)

Question. Evaluate: \(\int \frac{dx}{x(x^5 + 3)}\) 
Answer: \(\frac{1}{15} \log \left| \frac{x^5}{x^5 + 3} \right| + C\)

Question. Evaluate: \(\int x^2 \cdot \cos^{-1} x dx\) 
Answer: \(\frac{x^3}{3} \cos^{-1} x - \frac{1}{3}\sqrt{1-x^2} + \frac{1}{9}(1-x^2)^{3/2} + C\)

Question. Evaluate: \(\int \frac{dx}{x(x^3 + 8)}\) 
Answer: \(\frac{1}{24} \log \left| \frac{x^3}{x^3 + 8} \right| + C\)

Question. Evaluate: \(\int \frac{3x + 5}{\sqrt{x^2 - 8x + 7}} dx\) 
Answer: \(3\sqrt{x^2 - 8x + 7} + 17 \log |(x - 4) + \sqrt{x^2 - 8x + 7}| + C\)

Question. Evaluate: \(\int \frac{1 - x^2}{x(1 - 2x)} dx\) 
Answer: \(\frac{1}{2}x + \log |x| - \frac{3}{4} \log |2x - 1| + C\)

Question. Evaluate: \(\int \frac{(x + 2)}{\sqrt{(x - 2)(x - 3)}} dx\) 
Answer: \(\sqrt{x^2 - 5x + 6} + \frac{9}{2} \log \left| \left( x - \frac{5}{2} \right) + \sqrt{x^2 - 5x + 6} \right| + C\)

Question. Evaluate: \(\int_0^{\pi} \frac{x}{1 + \sin x} dx\) 
Answer: \(\pi\)

Question. Evaluate: \(\int_0^4 (|x| + |x - 2| + |x - 4|) dx\) 
Answer: 20

Question. Evaluate the following indefinite integral : \(\int \frac{\sin \phi}{\sqrt{\sin^2 \phi + 2 \cos \phi + 3}} d\phi\) 
Answer: \(-\sin^{-1} \left( \frac{\cos \phi - 1}{2} \right) + C\)

Question. Find : \(\int \frac{x^2}{x^4 + x^2 - 2} dx\) 
Answer: \(\frac{\sqrt{2}}{3} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) + \frac{1}{6} \log \left| \frac{x - 1}{x + 1} \right| + C\)

Question. Find: \(\int (3x + 1)\sqrt{4 - 3x - 2x^2} dx\)
Answer: \(-\frac{3}{4}(4-3x-2x^2)^{3/2} - \frac{5}{8}\left[ \frac{4x+3}{4}\sqrt{4-3x-2x^2} + \frac{41}{8} \sin^{-1} \left( \frac{4x+3}{\sqrt{41}} \right) \right] + C\)

Question. Evaluate : \(\int_0^{\pi} \frac{x \sin x}{1 + 3 \cos^2 x} dx\) 
Answer: \(\frac{\pi^2}{4\sqrt{3}}\)

Question. Find : \(\int \frac{x^2 + x + 1}{(x^2 + 1)(x + 2)} dx\) 
Answer: \(\frac{3}{5} \log |x + 2| + \frac{1}{5} \log |x^2 + 1| + \frac{1}{5} \tan^{-1} x + C\)

Question. Find : \(\int (x + 3)\sqrt{3 - 4x - x^2} dx\) 
Answer: \(-\frac{1}{3}(3-4x-x^2)^{3/2} + \frac{x+2}{2}\sqrt{3-4x-x^2} + \frac{7}{2} \sin^{-1} \left( \frac{x+2}{\sqrt{7}} \right) + C\)

Question. Find : \(\int \frac{(x^2 + 1)(x^2 + 4)}{(x^2 + 3)(x^2 - 5)} dx\)
Answer: \(x - \frac{\sqrt{3}}{4} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + \frac{9\sqrt{5}}{40} \log \left| \frac{x - \sqrt{5}}{x + \sqrt{5}} \right| + C\)

Question. Evaluate the following definite integral : \(\int_{-\pi}^{\pi} \frac{2x(1 + \sin x)}{1 + \cos^2 x} dx\) 
Answer: \(\pi^2\)

Question. Evaluate : \(\int_{-\pi/2}^{\pi/2} e^{2x} \left( \frac{1 - \sin 2x}{1 - \cos 2x} \right) dx\) 
Answer: \(-\frac{1}{2} e^x \cot x |_{-\pi/2}^{\pi/2}\)

Question. Evaluate : \(\int_0^{\pi/2} \log \sin x dx\)
Answer: \(-\frac{\pi}{2} \log 2\)

Question. Evaluate : \(\int \frac{2x^2 + 3}{x^2 + 5x + 6} dx\) 
Answer: \(2x - 11 \log |x + 2| + 21 \log |x + 3| + C\)

Question. Evaluate : \(\int e^{2x} \cdot \sin(3x + 1) dx\) 
Answer: \(\frac{e^{2x}}{13} [2 \sin(3x+1) - 3 \cos(3x+1)] + C\)

Question. Evaluate : \(\int \frac{1 - \cos x}{\cos x (1 + \cos x)} dx\) 
Answer: \(\log |\sec x + \tan x| - 2 \tan \frac{x}{2} + C\)

Question. Find: \(\int \frac{\sin 2x}{(\sin^2 x + 1)(\sin^2 x + 3)} dx\) 
Answer: \(\frac{1}{2} \log \left| \frac{\sin^2 x + 1}{\sin^2 x + 3} \right| + C\)

Long Answer Questions:

Question. Evaluate: \(\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\cot x}}\) 
Answer: \(\frac{\pi}{12}\)

Question. Evaluate \(\int_1^3 (e^{2 - 3x} + x^2 + 1) dx\) as a limit of a sum. 
Answer: \(\frac{-1}{3}(e^{-7} - e^{-1}) + \frac{32}{3}\)

Question. Evaluate: \(\int \frac{\sin x - x \cos x}{x(x + \sin x)} dx\) 
Answer: \(\log |x| - \log |x + \sin x| + C\)

Question. Find: \(\int_0^{\pi/4} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}}\)
Answer: \(\frac{5}{2}\)

Question. Evaluate : \(\int_0^{\pi/2} \frac{\cos^2 x dx}{1 + 3 \sin^2 x}\) 
Answer: \(\frac{\pi}{4}\)