Mathematics

# Evaluate the following definite integral:$\displaystyle \int _0^{\pi/2} \sin x \cos x dx$ is equal to

##### SOLUTION
$I=\displaystyle \int _0^{\pi/2} \sin x \cos x dx$

$\sin x=t\implies \cos x dx=dt\\x\to 0\to \dfrac \pi 2\\t\to 0\to 1$

$I=\displaystyle \int _0^{\pi/2} t dt$

$I=\left.\dfrac {t^2}2\right|^1_0$

$I=\dfrac 12-0$

$I=\dfrac 12$

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

#### Realted Questions

Q1 Subjective Medium
Solve: $\displaystyle \int \dfrac{(x^4 - x)^{1/4}}{x^5}dx.$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle \int \dfrac{x^4 + 1}{x^6 + 1} dx = \tan^{-1} (f(x)) -\dfrac{2}{3} \tan^{-1} (g(x)) + C$, then
• A. $g(x)$ is monotonic function
• B. none of these
• C. None
• D. Both $f(x)$ & $g(x)$ are odd functions

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium

Value of the integral $I=\displaystyle \int_{0}^{1}x(1-x)^{\mathrm{n}}dx=$
• A. $\displaystyle \frac{1}{n+2}$
• B. $\displaystyle \frac{1}{n+1}+\frac{1}{n+2}$
• C. $\displaystyle \frac{1}{n+1}$
• D. $\displaystyle \frac{1}{n+1}-\frac{1}{n+2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Solve: $\displaystyle \int _{ 0 }^{ \pi /4 }{ \tan ^{ 100 }{ x } } dx+\int _{ 0 }^{ \pi /4 }{ \tan ^{ 102 }{ x } } dx=....$
• A. $\cfrac { 1 }{ 102 }$
• B. $\cfrac { 1 }{ 100 }$
• C. $101$
• D. $\cfrac { 1 }{ 101 }$

Let $g(x)$ be a function defined on $[0, 7]$ and $g(x)=\int_0^x f(t) dt$, where $y=f(x)$ is the function whose graph is as shown in figure given below, then