MTH102: Elementary Mathematics Ii TMA3

MTH102: Elementary Mathematics Ii TMA3

Instructions: Answer All .
After Submission, you will not be able to answer these questions

Question 1 : Evaluate $$\int 3e^x+5\cos \left(x\right)-10{\sec }^2\left(x\right)dx$$

$$3e^x+5\sin x-10\sec x+c$$
$$2e^x-x-10\tan x+c$$
$$3e^x+\cos x-10\tan x+c$$
$$3e^2+5\sin x-10\tan x+c$$

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Question 2 : Find the $$\int {\tan }^{3}x{\sec }^{3}xdx$$
$${\cot }^{2}x+1$$
$${\tan }^{2}x+1$$
$${\sec }^{2}x$$
$${\sec }^{2}x+1$$

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Question 3 : Find $$\int {\sec }^{3}x\tan xdx$$
$$\frac{{\cos }^{3}x}{2}+c$$
$$\cos 2x+c$$
$$\sin 2x+c$$
$$\frac{{\sec }^{3}x}{3}+c$$

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Question 4 : Evaluate $$\int \frac{x+1}{x^2-3x+2}dx$$
$$3ln(x-2)+2ln(x+1)+c$$
$$3ln(x-2)-2ln(x-1)+c$$
$$-3ln(x-2)-2ln(x-1)+c$$
$$3ln(x+2)-2ln(x+1)+c$$

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Question 5 : Integrate with respect to x :$${\int }_{-1}^3\frac{x}{\sqrt{7+x^2}}dx$$
$$2\sqrt{2}$$
$$\sqrt{2}$$
$$4\sqrt{2}$$
$$4-2\sqrt{2}$$

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Question 6 : Evaluate $$\int (3x-2{)}^{6}dx$$
$$\frac{(3x+2{)}^7}{21}+c$$
$$\frac{(3x+2{)}^7}{2}+c$$
$$\frac{(3x-2{)}^7}{21}+c$$
$$\frac{3(3x-2{)}^7}{2}+c$$

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Question 7 : Integrate with respect to x :$${\int }_{-1}^2\frac{x^2}{(x^3+4{)}^2}dx$$
12
6
$$\frac{5}{12}$$
$$\frac{1}{2}$$

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Question 8 : Evaluate $$\int e^{4x}dx$$
$$3e^{\frac{x}{3}}+c$$
$$\frac{1}{3}ex+c$$
$$\frac{1}{4}{e}^{4x}+c$$
$$ex+c$$

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Question 9 : Given f(x) = $$(7x^4-5x^3)$$, evaluate $$\frac{df\left(x\right)}{dx}$$
$$28x^3-15x^2$$
$$28x^2-15x^2$$
$$2x^3-15x^2$$
$$7x^4-5x^3$$

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Question 10 : Find the integral with respect to x $$\int \cos x\sin xdx$$
$$\frac{\sin {}^{2}x}{2}+c$$
$$\sin x$$
$$\frac{{\cos }^{2}x}{2}+c$$
$$\sin 2x+c$$

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Question 11 : Integrate with respect to x :$${\int }_1^4\frac{x+1}{\sqrt{x}}dx$$
$$\frac{20}{3}$$
20
$$\frac{3}{20}$$
-20

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Question 12 : Determine $$\int \frac{x^2+1}{(x+2{)}^3}$$
$$-ln(x+2)-\frac{4}{x+2}-\frac{5}{2(x+3{)}^2}+c$$
$$ln(x-2)+\frac{4}{x-2}-\frac{5}{2(x+3{)}^2}+c$$
$$ln(x+2)-\frac{4}{x+2}-\frac{5}{2(x+3{)}^2}+c$$
$$ln(x+2)+\frac{4}{x+2}-\frac{5}{2(x+3{)}^2}+c$$

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Question 13 : Integrate $$\int (x^3+3x^2+2x+4)$$
$$6x^4-3x^2+c$$
$$3\frac{x^4}{4}+2x^3+x+c$$
$$\frac{x^4}{4}+x^3+x^2+4x+c$$
$$\frac{x^4}{2}-x^3+x^2+4x+c$$

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Question 14 : Evalute $$\int x^2(3-10x^3)dx$$
$$\frac{1}{15}(3-20x^3{)}^5)+c$$
$$\frac{1}{10}(1-10x^2{)}^5)+c$$
$$\frac{1}{100}(3-2x^3{)}^5)+c$$
$$\frac{1}{150}(3-10x^3{)}^5)+c$$

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Question 15 : Find $$\int x\cos a{x}^{2}dx$$ with respect to x
$$\sin 2x+c$$
$${\cos }^{3}x+c$$
$${\sec }^{2}x+1$$
$$\frac{1}{2}a\sin a{x}^2+c$$

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Question 16 : Evaluate $$\int \cos (6x+4)dx$$
$$\frac{\cos (6x+4)}{6}+c$$
$$\frac{\sin (6x+4)}{6}+c$$
$$\frac{\sec (6x+4)}{6}+c$$
$$\frac{\tan (6x+4)}{6}+c$$

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Question 17 : Evaluate $${\int }_{-1}^2{y}^2+y^{-2}dy$$
$$\frac{17}{16}$$
$$\frac{3}{16}$$
$$\frac{7}{16}$$
$$\frac{5}{16}$$

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Question 18 : Evaluate $$\int x{e}^{6x}dx$$
$$\frac{x}{6}{e}^{6x}-1\frac{1}{36}{e}^{6x}+c$$
$$\frac{x}{3}{e}^{6x}+1\frac{1}{6}{e}^{6x}+c$$
$$\frac{x}{6}{e}^{6x}+1\frac{1}{36}{e}^{6x}+c$$
$$-\frac{x}{6}{e}^{6x}+1\frac{1}{36}{e}^{6x}+c$$

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Question 19 : Find the volume of a sphere generated by a semicircle $$y=\sqrt{(}{r}^2-x^2)$$ revolving around the x-axis
$$4\pi \frac{r^3}{2}$$
$$-\pi \frac{-r^3}{2}$$
$$4\pi \frac{r^3}{3}$$
$$\pi \frac{r^3}{4}$$

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Question 20 : Evaluate $$\int x^2{e}^{3x}dx$$
$$\frac{e^x}{3}\left(x^2+\frac{2x}{3}-\frac{2}{5}\right)+c$$
$$-\frac{e^{3x}}{3}\left(x^2+\frac{2x}{3}-\frac{2}{9}\right)+c$$
$$\frac{e^{3x}}{3}\left(x^2-\frac{2x}{3}+\frac{2}{9}\right)+c$$
$$\frac{e^{2x}}{3}\left(x^3-\frac{x}{4}+\frac{2}{9}\right)+c$$