Mathematics

# Integrate:$\int _{ a-c }^{ b-c }{ f\left( x+c \right) } dx$ equals ?

$\int _{ b }^{ a }{ f\left( x+c \right) } dx$

##### SOLUTION

$Let\>x+c\>=\>t\\dx=dt\\at\>x=a-c,\>t=x+c=a\\at\>x=b-c,\>t=x+c=b\\I=\int\>_{\>a\>}^{\>b\>}f(t)\>dt\\=\int\>_{\>a\>}^{\>b\>}f(x+c)dx$

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

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int \frac{1}{\left ( x+1 \right )\sqrt{x^2-1}}dx$
• A. $\displaystyle \sqrt{\frac{x-1}{x^2+1}}$
• B. $\displaystyle \sqrt{\frac{x+1}{x-1}}+C$
• C. $\displaystyle \sqrt{\frac{x+1}{x-1}}$
• D. $\displaystyle \sqrt{\frac{x-1}{x+1}}+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Find the antiderivative of f(x) given by $f\left( x \right) =4{ x }^{ 3 }-\frac { 3 }{ { x }^{ 4 } }$ such that f(2)=0

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate $\displaystyle \int \frac{e^{x}-2}{e^{2x}+4}dx.$
• A. $-\displaystyle \dfrac{1}{2}x+log(e^{2x}+4)-\dfrac{1}{2}\tan ^{-1}\left ( \dfrac{e^{x}}{2} \right )$
• B. $-2x-log(e^{2x}+4)+\dfrac{1}{2}\tan ^{-1}\left ( \dfrac{e^{x}}{2} \right )$
• C. $-x+log(e^{2x}+4)+\dfrac{1}{2}\tan ^{-1}\left ( \dfrac{e^{x}}{2} \right )$
• D. $\displaystyle \dfrac{1}{2}x-log(e^{2x}+4)+\dfrac{1}{2}\tan ^{-1}\left ( \frac{e^{x}}{2} \right )$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $x$ satisfies the equation $\displaystyle\left(\int_0^1{\frac{dt}{t^2+2t\cos{\alpha}+1}}\right)x^2-\left(\int_{-3}^3{\frac{t^2\sin{2t}}{t^2+1}dt}\right)x-2=0$
for $(0<\alpha<\pi)$
then the value of $x$ is?
• A. $\displaystyle\pm\sqrt{\frac{\alpha}{2\sin{\alpha}}}$
• B. $\displaystyle\pm\sqrt{\frac{2\sin{\alpha}}{\alpha}}$
• C. $\displaystyle\pm\sqrt{\frac{\alpha}{\sin{\alpha}}}$
• D. $\displaystyle\pm2\sqrt{\frac{\sin{\alpha}}{\alpha}}$

$\int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx}$