Mathematics

# Evaluate the following definite integrals as limit of sums.$\displaystyle \int_{0}^{4}{(x+e^{2x})dx}$

##### SOLUTION
$\begin{array}{l} \int _{ 0 }^{ 4 }{ \left( { x+{ e^{ 2x } } } \right) dx } \\ \Rightarrow \int _{ 0 }^{ 4 }{ xdx } +\int _{ 0 }^{ 4 }{ { e^{ 2x } }dx } \\ \Rightarrow \dfrac { 1 }{ 2 } \left[ { { x^{ 2 } } } \right] _{ 0 }^{ 4 }+\dfrac { 1 }{ 2 } \left[ { { e^{ 2x } } } \right] _{ 0 }^{ 4 }+c \\ \Rightarrow \dfrac { 1 }{ 2 } \times 16=\dfrac { 1 }{ 2 } \left[ { { e^{ 8 } }-1 } \right] +c \\ 8+\dfrac { 1 }{ 2 } \left[ { { e^{ 8 } }-1 } \right] +c. \end{array}$

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

#### Realted Questions

Q1 Single Correct Medium
$\int _{ -\pi }^{ \pi }{ \cfrac { { cos }^{ 4 }x }{ { sin }^{ 4 }{ x+cos }^{ 4 }x } } dx$=...
• A. $\cfrac { \pi } {2}$
• B. $\pi$
• C. $2 \pi$
• D. $\cfrac { \pi} {4}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Show that $\displaystyle \int_{0}^{\dfrac{\pi}{2}} \dfrac{sin^2x}{sin \,x + cos \,x} = \dfrac{1}{\sqrt 2} log (\sqrt 2 + 1)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate$\int\limits_{} {\dfrac{{\cos x}}{{\left( {2 + \sin x} \right)\left( {3 + 4\sin x} \right)}}dx.}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle \int_{1/4}^{x}\displaystyle \frac{dt}{\sqrt{t-t^{2}}}= \displaystyle \frac{\pi }{6}$, then $x$ equals
• A. $1/3$
• B. $1$
• C. none of these
• D. $1/2$

Solve $\int {\sqrt {\dfrac{{1 + x}}{{1 - x}}} }$