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Fundamental theorems of integral calculus

Is the sentence true or false?
$\int_2^3v^2dv=\frac{19}{3}$

True

False

$\int_a^b\frac{d}{dx}\left[f\left(v\right)\right]dv=.........,\ a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.$

$\int_a^xf\left(v\right)dv$

$f\left(b\right)-f\left(a\right)$

$F\left(b\right)-F\left(a\right)$

Is the sentence true or false?
$\int_a^xf\left(v\right)dv=F\left(x\right),\ a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.$

True

False

Fill in the blanks: $\frac{d}{dx}\left[F\left(x\right)\right]=.......\ ,\ a\le x\le b,\ F\left(x\right)is\ the\ antiderivative\ of\ f.$

$\int_a^xf\left(v\right)dv$

$f\left(x\right)$

$F\left(x\right)$

Is the sentence true or false?
$\int_a^bf\left(v\right)dv=f\left(b\right)-f\left(a\right),\ a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.$

True

False

Fill in the blanks: $\frac{d}{dx}\left[\int_a^{h\left(x\right)}f\left(v\right)dv\right]$ =.........., $a\le x\le b,\ F\left(x\right)is\ the\ antiderivative\ of\ f.$

$\frac{d}{dx}\left[\int_a^{h\left(x\right)}f\left(v\right)dv\right]$

$f\left[h\left(x\right)\right]\ \ \frac{d}{dx}h\left(x\right)$

$F\left(h\left(x\right)\right)$

Fill in the blanks: $\int_a^bf\left(v\right)dv=...........,a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ off.$

$\int_a^xf\left(v\right)dv$

$f\left(b\right)-f\left(a\right)$

$F\left(b\right)-F\left(a\right)$

Fill in the blanks: $\int_a^bf\left(v\right)dv=..........,a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.$

$\int_a^xf\left(v\right)dv$

$f\left(b\right)-f\left(a\right)$

$F\left(b\right)-F\left(a\right)$

Is the sentence true or false?
$\int_4^8-2v^3dv=-1920$

True

False

Fill in the blanks:
$\frac{d}{dx}\left[F\left(x\right)\right]=...........,a\le x\le b$ , F(x) is the antiderivative of f.

$\int_a^xf\left(v\right)dv$

$f\left(x\right)$

$F\left(x\right)$

Questions

Time

Score

Fundamental theorems of integral calculus

Is the sentence true or false?﻿﻿∫23v2dv=193\int_2^3v^2dv=\frac{19}{3}∫23​v2dv=319​﻿

True

False

﻿∫abddx[f(v)]dv=........., a≤x≤b, F(x) is the antiderivative of f.\int_a^b\frac{d}{dx}\left[f\left(v\right)\right]dv=.........,\ a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.∫ab​dxd​[f(v)]dv=........., a≤x≤b, F(x) is the antiderivative of f.﻿

﻿∫axf(v)dv\int_a^xf\left(v\right)dv∫ax​f(v)dv﻿

﻿f(b)−f(a)f\left(b\right)-f\left(a\right)f(b)−f(a)﻿

﻿F(b)−F(a)F\left(b\right)-F\left(a\right)F(b)−F(a)﻿

Is the sentence true or false?﻿∫axf(v)dv=F(x), a≤x≤b, F(x) is the antiderivative of f.\int_a^xf\left(v\right)dv=F\left(x\right),\ a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.∫ax​f(v)dv=F(x), a≤x≤b, F(x) is the antiderivative of f.﻿

True

False

Fill in the blanks:﻿ddx[F(x)]=....... , a≤x≤b, F(x)is the antiderivative of f.\frac{d}{dx}\left[F\left(x\right)\right]=.......\ ,\ a\le x\le b,\ F\left(x\right)is\ the\ antiderivative\ of\ f.dxd​[F(x)]=....... , a≤x≤b, F(x)is the antiderivative of f.﻿

﻿∫axf(v)dv\int_a^xf\left(v\right)dv∫ax​f(v)dv﻿

﻿f(x)f\left(x\right)f(x)﻿

﻿F(x)F\left(x\right)F(x)﻿

True

False

Fill in the blanks:﻿ddx[∫ah(x)f(v)dv]\frac{d}{dx}\left[\int_a^{h\left(x\right)}f\left(v\right)dv\right]dxd​[∫ah(x)​f(v)dv]﻿ =..........,﻿a≤x≤b, F(x)is the antiderivative of f.a\le x\le b,\ F\left(x\right)is\ the\ antiderivative\ of\ f.a≤x≤b, F(x)is the antiderivative of f.﻿

﻿ddx[∫ah(x)f(v)dv]\frac{d}{dx}\left[\int_a^{h\left(x\right)}f\left(v\right)dv\right]dxd​[∫ah(x)​f(v)dv]﻿

﻿f[h(x)] ddxh(x)f\left[h\left(x\right)\right]\ \ \frac{d}{dx}h\left(x\right)f[h(x)] dxd​h(x)﻿

﻿F(h(x))F\left(h\left(x\right)\right)F(h(x))﻿

Fill in the blanks: ﻿∫abf(v)dv=...........,a≤x≤b, F(x) is the antiderivative off.\int_a^bf\left(v\right)dv=...........,a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ off.∫ab​f(v)dv=...........,a≤x≤b, F(x) is the antiderivative off.﻿

﻿∫axf(v)dv\int_a^xf\left(v\right)dv∫ax​f(v)dv﻿

﻿f(b)−f(a)f\left(b\right)-f\left(a\right)f(b)−f(a)﻿

﻿F(b)−F(a)F\left(b\right)-F\left(a\right)F(b)−F(a)﻿

Fill in the blanks: ﻿∫abf(v)dv=..........,a≤x≤b, F(x) is the antiderivative of f.\int_a^bf\left(v\right)dv=..........,a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.∫ab​f(v)dv=..........,a≤x≤b, F(x) is the antiderivative of f.﻿

﻿∫axf(v)dv\int_a^xf\left(v\right)dv∫ax​f(v)dv﻿

﻿f(b)−f(a)f\left(b\right)-f\left(a\right)f(b)−f(a)﻿

﻿F(b)−F(a)F\left(b\right)-F\left(a\right)F(b)−F(a)﻿

Is the sentence true or false?﻿﻿∫48−2v3dv=−1920\int_4^8-2v^3dv=-1920∫48​−2v3dv=−1920﻿

True

False

Fill in the blanks: ﻿ddx[F(x)]=...........,a≤x≤b\frac{d}{dx}\left[F\left(x\right)\right]=...........,a\le x\le bdxd​[F(x)]=...........,a≤x≤b﻿, F(x) is the antiderivative of f.

﻿∫axf(v)dv\int_a^xf\left(v\right)dv∫ax​f(v)dv﻿

﻿f(x)f\left(x\right)f(x)﻿

﻿F(x)F\left(x\right)F(x)﻿

Incorrect

Great!

Is the sentence true or false?
$\int_2^3v^2dv=\frac{19}{3}$

$\int_a^b\frac{d}{dx}\left[f\left(v\right)\right]dv=.........,\ a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.$

$\int_a^xf\left(v\right)dv$

$f\left(b\right)-f\left(a\right)$

$F\left(b\right)-F\left(a\right)$

Is the sentence true or false?
$\int_a^xf\left(v\right)dv=F\left(x\right),\ a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.$

Fill in the blanks: $\frac{d}{dx}\left[F\left(x\right)\right]=.......\ ,\ a\le x\le b,\ F\left(x\right)is\ the\ antiderivative\ of\ f.$

$\int_a^xf\left(v\right)dv$

$f\left(x\right)$

$F\left(x\right)$

Fill in the blanks: $\frac{d}{dx}\left[\int_a^{h\left(x\right)}f\left(v\right)dv\right]$ =.........., $a\le x\le b,\ F\left(x\right)is\ the\ antiderivative\ of\ f.$

$\frac{d}{dx}\left[\int_a^{h\left(x\right)}f\left(v\right)dv\right]$

$f\left[h\left(x\right)\right]\ \ \frac{d}{dx}h\left(x\right)$

$F\left(h\left(x\right)\right)$

Fill in the blanks: $\int_a^bf\left(v\right)dv=...........,a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ off.$

$\int_a^xf\left(v\right)dv$

$f\left(b\right)-f\left(a\right)$

$F\left(b\right)-F\left(a\right)$

Fill in the blanks: $\int_a^bf\left(v\right)dv=..........,a\le x\le b,\ F\left(x\right)\ is\ the\ antiderivative\ of\ f.$

$\int_a^xf\left(v\right)dv$

$f\left(b\right)-f\left(a\right)$

$F\left(b\right)-F\left(a\right)$

Is the sentence true or false?
$\int_4^8-2v^3dv=-1920$

Fill in the blanks:
$\frac{d}{dx}\left[F\left(x\right)\right]=...........,a\le x\le b$ , F(x) is the antiderivative of f.

$\int_a^xf\left(v\right)dv$

$f\left(x\right)$

$F\left(x\right)$