Mathematics

# $\displaystyle \int\dfrac {\cos 2x+2\sin^{2}x}{\cos^{2}x}dx$

##### SOLUTION

Consider the following question.

$I=\int_{{}}^{{}}{\dfrac{\cos 2x+2{{\sin }^{2}}x}{{{\cos }^{2}}x}}dx$

$=\int_{{}}^{{}}{\dfrac{{{\cos }^{2}}x-{{\sin }^{2}}x+2{{\sin }^{2}}x}{{{\cos }^{2}}x}}dx$

$=\int_{{}}^{{}}{\dfrac{{{\sin }^{2}}x+{{\cos }^{2}}x}{{{\cos }^{2}}x}}dx$

$=\int_{{}}^{{}}{{{\sec }^{2}}xdx}$

$=\tan x+C$

Hence, this is the required answer.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
If $\displaystyle I_1=\int_{0}^{\frac{\pi}{2}}e^{-x} sin 4x dx$  and $\displaystyle I_2=\int_0^{2\pi}e^{-x} sin 4x dx$ and  $\displaystyle I_2=\lambda I_1$, then $\displaystyle \lambda$ is equal to
• A. $\displaystyle \frac{e^{2\pi}-1}{e^{\pi}-1}$
• B. $\displaystyle \frac{e^{\pi}-1}{1-e^{2\pi}}$
• C. $\displaystyle \frac{1-e^{2\pi}}{1-e^{\frac{\pi}{2}}}$
• D. $\displaystyle \frac{1-e^{-2\pi}}{1-e^{-\frac{\pi}{2}}}$

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Q2 Subjective Medium
Show that $\int \displaystyle x\sin x\cos x = f(x)$, taking const. of integration  as zero. Find $f(\pi /4)$

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Q3 Subjective Medium
$\displaystyle\int_{0}^{\pi/4}\dfrac{\tan^{3}x}{1+\cos 2x}dx$

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Q4 Single Correct Medium
Solve :
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Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$