Question

just want to check my answers

Answer 1

\[If F(x)=\int\limits_{x}^{x^2}\frac{ 1 }{ t^2+1 }dt--> F'(x)=\frac{ 2x }{ x ^4+1}\]

Answer 2

\[If F(x)=\int\limits_{x}^{1}\sqrt{t^2+1}dt--> then F'(x)=\sqrt{2}-\sqrt{x^2+1}\]

Answer 3

\(\large \color{red}{\checkmark}\)
\(\large \color{red}{\checkmark}\)

Answer 4

thank u

Answer 5

\[If F(x)=\int\limits_{x}^{2x}\frac{ 1 }{ t}dt-->for x>0, then F'(x)=\frac{ 1 }{ x }\]

Answer 6

\[If F(x)=\int\limits_{\pi}^{x^2}\frac{ \sin(t) }{t}--> for x>0, then F'(x)=\frac{ 2\sin(x^2) }{ x }\]

Answer 7

\(If F(x)=\int\limits_{x}^{2x}\frac{ 1 }{ t}dt-->for x>0, then F'(x)=\color{red}{0}\)

Answer 8

wait can u explain this

Answer 9

\(\large \frac{d}{dx}\Big(\int\limits_{x}^{2x}\frac{ 1 }{ t}dt\Big) = \frac{d}{dx}\Big(\int\limits_{x}^{0}\frac{ 1 }{ t}dt\Big) + \frac{d}{dx}\Big(\int\limits_{0}^{2x}\frac{ 1 }{ t}dt\Big) \)

Answer 10

\(\large = \frac{d}{dx}\Big(-\int\limits_{0}^{x}\frac{ 1 }{ t}dt\Big) + \frac{d}{dx}\Big(\int\limits_{0}^{2x}\frac{ 1 }{ t}dt\Big) \)

\(\large = -\frac{ 1 }{ x} + \frac{1}{2x}*2\)

\(\large = -\frac{ 1 }{ x} + \frac{1}{x}\)

\(\large = 0\)

Answer 11

ok so we replace the interval on the top to 0 and add to the original

Answer 12

and for the second one is the same thing right

Answer 13

let me see 2nd one :)

Answer 14

one sec, lets have a look at below :
\(If F(x)=\int\limits_{x}^{1}\sqrt{t^2+1}dt--> then F'(x)=\sqrt{2}-\sqrt{x^2+1} \)

Answer 15

earlier, i overlooked sorry.
answer is wrong for this one

Answer 16

you should get just this :

\(If F(x)=\int\limits_{x}^{1}\sqrt{t^2+1}dt--> then F'(x)=-\sqrt{x^2+1} \)

Answer 17

ok so its just negative

Answer 18

\(\large \frac{d}{dx}\Big(\int\limits_{x}^{1}\sqrt{t^2+1}dt\Big) = \frac{d}{dx}\Big(\int\limits_{x}^{0}\sqrt{t^2+1}dt\Big) + \frac{d}{dx}\Big(\int\limits_{0}^{1}\sqrt{t^2+1}dt\Big) \)

\(\large \frac{d}{dx}\Big(\int\limits_{x}^{0}\sqrt{t^2+1}dt\Big) + 0\)

\(\large -\sqrt{x^2+1}\)

Answer 19

yes

Answer 20

ok and the last one please :)

Answer 21

the second derivative of integral becomes 0
cuz derivative of constant is 0

Answer 22

yeah 1 sec let me see

Answer 23

last one is \(\large \color{red}{\checkmark}\)

Answer 24

thank u so much for helping me :)

Answer 25

np :)