2 Marks Questions

Question 1512 Marks

Evaluate $\int(1-\text{x})\sqrt{\text{x}}\text{ dx}$

Answer

Let $\sqrt{\text{x}}=\text{t}$
$\text{x}=\text{t}^2$
$1-\text{x}=1-\text{t}^2$
$-\text{dx}=-2\text{tdt}$
$\text{dx}=2\text{tdt}$
$\text{I}=\int(1-\text{x})\sqrt{\text{x}}\text{ dx}$
$=\int(1-\text{t}^2)\text{t}2\text{t dt}$
$=2\int (1-\text{t}^2)\text{t}^2\text{ dt}$
$=2\big(\int\text{t}^2\text{dt}-\int\text{t}^4\text{dt}\big)$
$=2\frac{\text{t}^3}{3}-2\frac{\text{t}^5}{5}+\text{C}$
$=\frac{2}{3}\text{x}^{\frac{3}{2}}-\frac{2}{5}\text{x}^{\frac{5}{2}}+\text{C}$

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Question 1522 Marks

Evaluate the integral in Exercise:$\int\limits^{\frac{\pi}{2}}_{0}\frac{\sin\text{x}}{1+\cos^{2}\text{x}}\text{dx}$

Answer

$\text{Let}\ \text{I}=\int\limits^{\frac{\pi}{2}}_{0}\frac{\sin\text{x}}{1+\cos^{2}\text{x}}\text{dx}$
$\text{put}\ \cos\text{x}=\text{t},\therefore\sin\text{x}\ \text{dx}=\text{dt}\Rightarrow\sin\text{x}\ \text{dx}=-\text{dt}\ \text{when}\ \text{x}=0,\text{t}=\cos0=1$
$\text{when}\ \text{x}=\frac{\pi}{2},\text{t}=\cos\frac{\pi}{2}=0$
$\therefore\ \ \text{I}=-\int\limits^{0}_{1}\frac{\text{dt}}{1+\text{t}^{2}}=-\big[\tan^{-1}\text{t}\big]^{0}_{1}=-\big[\tan^{-1}0-\tan^{-1}1\big]=-\bigg[0-\frac{\pi}{4}\bigg]=\frac{\pi}{4}$

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Question 1532 Marks

$\int\cos^2\text{nx dx}$

Answer

$\int\cos^2\text{nx dx}$
$=\int\Big[\frac{1+\cos2\text{nx}}{2}\Big]\text{dx}$ $\Big[\therefore\cos^2\text{x}=\frac{1+\cos2\text{x}}{2}\Big]$
$=\frac{1}{2}\int(1+\cos2\text{nx})\text{dx}$
$=\frac{1}{2}\Big[\text{x}+\frac{\sin2\text{nx}}{2\text{n}}\Big]+\text{C}$
$=\frac{\text{x}}{2}+\frac{\sin2\text{nx}}{4\text{n}}+\text{C}$

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Question 1542 Marks

Integrate the functions in Exercises:
$\frac{1}{\sqrt{(2-\text{x})^2}+1}$

Answer

$\int\frac{1}{\sqrt{(2-\text{x})^2}+1}\text{ dx}$
$=\frac{\log\bigg|(2-\text{x})+\sqrt{(2-\text{x})^2+1^2}\bigg|}{-1\rightarrow\text{Coeff. of x}}+\text{c}$$ \ \ \ \ \ \ \ \bigg[\because\int\frac{1}{\sqrt{\text{x}^2+\text{a} ^2}}\text{ dx}=\log\bigg|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\bigg|\bigg]$
$=-\log\bigg|2-\text{x}+\sqrt{4+\text{x}^2-4\text{x}+1}\bigg|+\text{c}$
$=\log\Bigg|\frac{1}{2-\text{x}\sqrt{\text{x}^2-4\text{x}+5}}\Bigg|+\text{C}$

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Question 1552 Marks

Evaluate the following integrals:
$\int\limits^3_2\frac{1}{\text{x}}\text{ dx}$

Answer

$\int\limits^3_2\frac{1}{\text{x}}\text{ dx}$
$=\Big[\log_\text{e}\text{x}\Big]^3_2$
$=\log_\text{e}3-\log_\text{e}2$
$=\log_\text{e}\Big(\frac{3}{2}\Big)$

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Question 1562 Marks

Evaluate the following integrals:$\int\text{xe}^{2\text{x}}\text{dx}$

Answer

Let $\text{I}=\int\text{xe}^{2\text{x}}\text{dx}$
Using integration by parts,
$\text{I}=\text{x}\int\text{e}^{2\text{x}}\text{dx}-\int(1\times\int\text{e}^{2\text{x}}\text{dx})\text{dx+C}$
$=\frac{\text{xe}^{2\text{x}}}{2}-\int\Big(\frac{\text{e}^{2\text{x}}}{2}\Big)\text{dx+C}$
$=\frac{\text{xe}^{2\text{x}}}{2}-\frac{\text{e}^{2\text{x}}}{4}+\text{C}$
$\text{I}=\Big(\frac{\text{x}}{2}-\frac{1}{4}\Big)\text{e}^{2\text{x}}+\text{C}$

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Question 1572 Marks

Integrate the function in Exercise:$\text{f}\big(\text{ax}+\text{b}\big)\big[\text{f(ax}+\text{b)}\big]^{\text{n}}$

Answer

$\text{f' (ax}+\text{b)}\big[\text{f (ax}+\text{b)}\big]^{\text{n}}$ $\text{Let}\ \text{f (ax}+\text{b)}=\text{t}\Rightarrow \text{af}\ \text{(ax}+\text{b)}\text{dx}=\text{dt}$ $\Rightarrow\int\text{f}\ \big(\text{ax}+\text{b}\big)\big[\text{f}\text{(ax}+\text{b)}\big]^{\text{n}}\text{dx}=\frac{1}{\text{a}}\int\text{t}^{\text{n}}\text{dt}$$=\frac{1}{\text{a}}\bigg[\frac{\text{t}^{\text{n}+1}}{\text{n}+1}\bigg]$
$=\frac{1}{\text{a}(\text{n}+1)}\big(\text{f (ax}+\text{b)}\big)^{\text{n+1}}+\text{C}$

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Question 1582 Marks

Evaluate the following integrals:
$\int\frac{1}{3\sqrt{\text{x}^2}}\text{dx}$

Answer

$\int\frac{\text{dx}}{3\sqrt{\text{x}^2}}$
$=\int\frac{\text{dx}}{\text{x}^\frac{2}{3}}$
$=\int\text{x}^\frac{-2}{3}\text{dx}$
$=\frac{\text{x}^{-\frac{2}{3}+1}}{-\frac{2}{3}+1}+\text{c}$
$=3\text{x}^\frac{1}{3}+\text{c}$

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Question 1592 Marks

Evaluate the following integrals:
$\int\frac{(\sin^{-1}\text{x})^3}{\sqrt{1-\text{x}^2}}=\text{dx}$

Answer

$\int\frac{(\sin^{-1}\text{x})^3}{\sqrt{1-\text{x}^2}}=\text{dx}$ Let $\sin^{-1}\text{x}=\text{t}$ $\Rightarrow\frac{1}{\sqrt{1-\text{x}^2}}\text{ dx}=\text{dt}$ Now, $\int\frac{(\sin^{-1}\text{x})^3}{\sqrt{1-\text{x}^2}}=\text{dx}$$=\int\text{t}^3\text{ dt}$
$=\frac{\text{t}^4}{4}+\text{C}$
$=\frac{(\sin^{-1}\text{x})^4}{4}+\text{C}$

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Question 1602 Marks

Evaluate the following integrals:
$\int\limits^4_0\frac{1}{\sqrt{16-\text{x}^2}}\text{ dx}$

Answer

$\int\limits^4_0\frac{1}{\sqrt{16-\text{x}^2}}\text{ dx}$
$=\int\limits^4_0\frac{1}{\sqrt{14^2-\text{x}^2}}\text{ dx}$
$=\Big[\sin^{-1}\frac{\text{x}}{4}\Big]^4_0$
$=\Big(\frac{\pi}{2}-0\Big)$
$=\frac{\pi}{2}$

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Question 1612 Marks

Evaluate the following definite integrals:
$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$

Answer

We know that $\int\frac{\text{dx}}{\text{x}}=\log\text{x}+\text{C}$
Now,
$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$
$=\big[\log(\text{x}+7)\big]^3_{-2}$
$=\big[\log10-\log5]^3_{-2}$
$=\log\frac{10}{5}$ $\Big[\because\log\text{a}-\log\text{b}=\log\frac{\text{a}}{\text{b}}\Big]$
$=\log2$
$\therefore\ \int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}=\log2$

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Question 1622 Marks

Evaluate the following integrals:
$\int3^{2\log_3\text{x}}\text{dx}$

Answer

$\int3^{2\log_3\text{x}}\text{dx}=\int3^{\log_3\text{x}^2}\text{dx}$
$=\int\text{x}^2\text{dx}$
$=\frac{\text{x}^3}{3}+\text{c}$
$\int\log_\text{x}\text{xdx}=\int1\text{dx}$
$=\text{x}+\text{c}$

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Question 1632 Marks

Write the primitive or anti-derivative of $\text{f(x)}=\sqrt{\text{x}}=\frac{1}{\sqrt{\text{x}}}$.

Answer

$\text{f(x)}=\sqrt{\text{x}}=\frac{1}{\sqrt{\text{x}}}$
Integrating both sides:
$\int\text{f(x)}\text{dx}=\int\Big(\sqrt{\text{x}}=\frac{1}{\sqrt{\text{x}}}\Big)\text{dx}$
$=\int\Big(\text{x}^{\frac{1}{2}}+\text{x}^{-\frac{1}{2}}\Big)\text{dx}$
$=\bigg[\frac{\text{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\bigg]+\bigg[\frac{\text{x}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\bigg]+\text{C}$
$=\frac{2}{3}\text{x}^{\frac{3}{2}}+2\text{x}^{\frac{1}{2}}+\text{C}$

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MATHS 2 Marks Questions