Mathematics

# Evaluate: $\int x \sin 2 x d x$

##### SOLUTION
$I=\int x\sin 2x$

We know the formula (by Parts)
$\displaystyle\int uv=u\int v-\int \left(\int v\right)\dfrac{du}{dn}dx$

$u=x;v\sin 2x$

$I=x\displaystyle \int \sin 2x-\int \left(\int \sin 2x\right)\dfrac{d}{dx}(x)dx$

$I=x\dfrac{(-\cos (2x))}{2}-\displaystyle\int \left(\dfrac{-\cos 2x}{2}\right)1dn$

$I=\dfrac{-x\cos 2x}{2}+\dfrac{\sin 2x}{2.2}+C$

$I=\dfrac{\sin 2x}{4}-\dfrac{x\cos 2x}{2}+C$.

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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

$\displaystyle \int_{1}^{2}\frac{\mathrm{d}\mathrm{x}}{\sqrt{1+\mathrm{x}^{2}}}=$
• A. $\displaystyle \log_{\mathrm{e}}(\frac{\sqrt{2}+1}{2+\sqrt{5}})$
• B. $\displaystyle \log_{\mathrm{e}}(\frac{2-\sqrt{5}}{\sqrt{2}-1})$
• C.
• D. $\displaystyle \log_{\mathrm{e}}(\frac{2+\sqrt{5}}{\sqrt{2}+1})$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Find :
$\int { \dfrac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int _{ 0 }^{ 2 }{ \dfrac { \sqrt { 2+x } }{ \sqrt { 2-x } } } dx=$
• A. ${\pi}+\dfrac{3}{2}$
• B. ${\pi}+1$
• C. $None\ of\ these$
• D. ${\pi}+2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve:
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Given that for each $\displaystyle a \in (0, 1), \lim_{h \rightarrow 0^+} \int_h^{1-h} t^{-a} (1 -t)^{a-1}dt$ exists. Let this limit be $g(a)$
In addition, it is given that the function $g(a)$ is differentiable on $(0, 1)$