Mathematics

# Evaluate the following definite integral:$\displaystyle\int_{0}^{\pi/2} \sin x \cos x\ dx$

##### SOLUTION

Consider, $I=\displaystyle\int_{0}^{\pi/2} \sin x \cos x\ dx$

Let $t=\sin x \implies dt=\cos x dx$

$x \to 0 \to \pi/2$ and $t \to 0 \to 1$

$I=\displaystyle \int _0^1 t \ dt$

$I= \left[\dfrac {t^2}2 \right] _0^1 =\dfrac 12$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
The value of the integral $\displaystyle\int\dfrac{\cos^{3}x+\cos^{5}x}{\sin^{2}x+\sin^{4}x}dx$ is
• A. $\sin x-6\tan^{-1}(\sin x)+c$
• B. $\sin x-2(\sin x)^{-1}+c$
• C. $\sin x-2(\sin x)^{-1}+5\tan^{-1}(\sin x)+c$
• D. $\sin x-2(\sin x)^{-1}-6\tan^{-1}(\sin x)+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int {\frac{x^2}{(x\,\sin\,x+\cos\,x)^2}} dx$ is equal to
• A. $\displaystyle \frac{\sin\,x+\cos\,x}{x\,\sin\,x+\cos\,x}+C$
• B. $\displaystyle \frac{x\,\sin\,x-\cos\,x}{x\,\sin\,x+\cos\,x}+C$
• C. None of these
• D. $\displaystyle \frac{\sin\,x-x\,\cos\,x}{x\,\sin\,x+\cos\,x}+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
lf $I_{n}=\displaystyle \int(\log x)^{n}dx$, then $I_{6}+6I_{5}=$
• A. $x(\log x)^{5}$
• B. $-x(\log x)^{5}$
• C. $-x(\log x)^{6}$
• D. $x(\log x)^{6}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve: $\displaystyle\int { \sqrt { 4-{ x }^{ 2 } } dx }$

Let $\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3} dx$  &  $\displaystyle I_{2}=\int_{0}^{1}(1-x^{3})^{1/2} dx$