Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\tan x \sec ^{2} x}{\left(1-\tan ^{2} x\right)} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{(1+\cot x)}{(x+\log \sin x)} d x$ Solution:...
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Question: Evaluate the following integrals: $\int\left(\frac{1-\tan x}{x+\log \cos x}\right) d x$ Solution:...
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Question: Evaluate the following integrals: $\left(\frac{1+\sin 2 x}{x+\sin ^{2} x}\right) d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{\sec ^{2}\left(2 \tan ^{-1} x\right)}{\left(1+x^{2}\right)} d x$ Solution:...
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Question: Evaluate the following integrals: Solution:...
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Question: Evaluate the following integrals: $\int \sqrt{e^{x}-1} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{e^{\sqrt{x}} \cos \left(e^{\sqrt{x}}\right)}{\sqrt{x}} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{(x+1) e^{x}}{\cos ^{2}\left(x e^{x}\right)} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{e^{m \tan ^{-1} x}}{\left(1+x^{2}\right)} d x$ Solution:...
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Question: Evaluate the following integrals: $\int x^{2} e^{x^{3}} \cos \left(e^{x^{3}}\right) d x$ Solution:...
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Question: Evaluate the following integrals: $\int \sqrt{1+x} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{d x}{(3-5 x)}$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{\sin 2 x}{(a+b \cos x)^{2}} d x$ Solution:...
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Question: Evaluate the following integrals: $\int 2 x \sec ^{3}\left(x^{2}+3\right) \tan \left(x^{2}+3\right) d x$ Solution:...
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Question: Evaluate the following integrals: $\int e^{-x} \operatorname{cosec}^{2}\left(2 e^{-x}+5\right) d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{\sec ^{2} x}{\sqrt{1-\tan ^{2} x}} d x$ Solution:...
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Question: Evaluate the following integrals: $\int x \sin ^{3} x^{2} \cos x^{2} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{(x+1)(x+\log x)^{2}}{x} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{(\cos x-\sin x)}{(1+\sin 2 x)} d x$ Solution:...
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Question: Evaluate the following integrals: $\int \frac{\cos 2 x}{(\sin x+\cos x)} d x$ Solution:...
Read More →Water level is maintained in a cylindrical vessel upto a fixed
Question: Water level is maintained in a cylindrical vessel upto a fixed height $\mathrm{H}$. The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel so that the water stream coming out of the hole strikes the horizontal plane at the greatest distance from the vessel.(figure 13-11) Solution:...
Read More →Water leaks out from an open tank through a hole of area
Question: Water leaks out from an open tank through a hole of area $2 \mathrm{~mm}^{2}$ in the bottom. Suppose water is filled up to a height of 80 $\mathrm{cm}$ and the area of cross-section of the tank is $0.4 \mathrm{~m}^{2}$. The pressure at the open surface and at the hole are equal to the atmospheric pressure. Neglect the small velocity of the water near the open surface in the tank. (a) Find the initial speed of water coming out of the hole. (b) Find the speed of water coming out when hal...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{d x}{(1-\cot x)}$ Solution:...
Read More →Water flows through the tube shown in figure(13-E10).
Question: Water flows through the tube shown in figure(13-E10). The areas of cross-section of the wide and the narrow portions of the tube are $5 \mathrm{~cm}^{2}$ and $2 \mathrm{~cm}^{2}$ respectively. The rate of flow of water through the tube is $500 \mathrm{~cm}^{3} / \mathrm{s}$. Find the difference of mercury levels in the U-tube. Solution:...
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