Mathematics

# Evaluate : $\int\limits_0^{\pi /2} {\dfrac{{\cos x\,dx}}{{\left( {\,\cos \,x\, + \,\sin x} \right)}}}$

##### SOLUTION
We have,
$I=\int\limits_0^{\pi /2} {\dfrac{{\cos x\,dx}}{{\left( {\,\cos \,x\, + \,\sin x} \right)}}}$              $.........(1)$

We know that
$\int\limits_b^{a}f(x)dx=\int \limits_b^a\ f(a+b-x)dx$

Therefore,
$I=\int\limits_0^{\pi /2} {\dfrac{{\sin x\,dx}}{{\left( {\,\sin \,x\, + \,\cos x} \right)}}}$             $............(2)$

On adding equations $(1)$ and $(2)$, we get
$2I=\int\limits_0^{\pi /2} {\dfrac{{\sin x+\cos x}}{{\left( {\,\sin \,x\, + \,\cos x} \right)}}} dx$

$2I=\int\limits_0^{\pi /2} 1 dx$

$2I=[x]_0^\frac{\pi}{2}$

$2I=\dfrac{\pi}{2}-0$

$2I=\dfrac{\pi}{2}$

$I=\dfrac{\pi}{4}$

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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 Medium
Solve $\int { (\log_ e x)^2} dx$ o
• A. $x \log_e x (\log_e x +2)+c$
• B. $x \log_e(2\log_e \ x + 1)+c$
• C. $x\{(\log_e x )^2 - 2(log_ex-2)\}+c$
• D. $x[(\log x)^2 - 2(\log x)+2] + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int_{0}^{\infty}{\dfrac{\sin^{2} x}{x^{2}}dx}$ must be same as
• A. $(\int_{0}^{\infty}{\dfrac{\sin x}{x}dx})^{2}$
• B. $\int_{0}^{\infty}{\dfrac{\cos^{2} x}{x^{2}}dx}$
• C. $none\ of\ these$
• D. $\int_{0}^{\infty}{\dfrac{\sin x}{x}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
What is $\int_{\frac{\pi}{2}}^{0} \ln (\tan x)dx$ equal to ?
• A. $\ln2$
• B. $-\ln2$
• C. None of the above
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
I : Number of partial fractions of $\displaystyle \frac{x^{3}+x^{2}+1}{x^{4}+x^{2}+1}$ is 4
II : Number of partial fractions of $\displaystyle \frac{3x+5}{(x-1)^{2}(x^{2}+1)^{3}}$ is 5

Which of the above statement is true.
• A. only I
• B. Both I and II
• C. Neither I nor II
• D. Only II

Solve $\displaystyle\int\limits_0^x {\dfrac{{\sqrt x }}{{\sqrt x + \sqrt {8 - x} }}dx}$