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Wize University Calculus 1 Textbook > Integrals

The Fundamental Theorem of Calculus - Part 1

Computing Definite Integrals GeometricallyPrevious Section
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Fundamental Theorem of Calculus (FTC) - Part I

The first part of The Fundamental Theorem of Calculus explores the direct relationship between derivatives and integrals.

Fundamental Theorem of Calculus: 1

Let f(x)f(x) be a continuous function. Define
F(x)=∫cxf(t)dt\boxed{\quad F(x)=\int_{c}^{x}f(t)\text{d}t\quad }

Then F(x)F(x) is differentiable and it is the antiderivative of f(x)f(x) , that is, F′(x)=f(x)F^{\prime}(x)=f(x).

ddx∫cxf(t) dt=f(x)\boxed{\quad \displaystyle\frac{d}{dx}\int_{c}^{x}f(t)\,\text{d}t=f(x)\quad }

Wize Concept
This means that integration and differentiation are "inverse" operations: if we apply one and then the other, the original function remains unchanged (up to addition of a constant).

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FTC Part 1 & the Chain Rule

If the bounds are functions of xx, we have (by the Chain Rule):

ddx∫cg(x)f(t) dt=f(g(x))⋅g′(x)\boxed{\quad \displaystyle\frac{d}{dx}\int_{c}^{g(x)}f(t)\,\text{d}t=f(g(x)) \cdot g'(x)\quad }

ddx∫h(x)g(x)f(t) dt=f(g(x))⋅g′(x)−f(h(x))⋅h′(x)\boxed{\quad \displaystyle\frac{d}{dx}\int_{h(x)}^{g(x)}f(t)\,\text{d}t=f(g(x)) \cdot g'(x)-f(h(x)) \cdot h'(x)\quad }

Wize Tip
When differentiating a definite integral, there is no need to antiderive the function! Simply substitute the bounds for tt and multiply by the derivatives of the bounds if necessary.

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Example: FTC Part 1

Find the derivative using the Fundamental Theorem of Calculus.

ddx[∫x23t−1t5+1dt]\displaystyle \frac{d}{dx}\left[\int_x^2\frac{3t-1}{t^{5}+1}dt\right]


=−ddx∫2x3t−1t5+1dt=\displaystyle -\frac{d}{dx}\int_2^x\frac{3t-1}{t^5+1}dt (adjust integration limits using the property of definite integral before applying FTC1)

=−3x−1x5+1=-\dfrac{3x-1}{x^5+1} (by FTC 1)
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Example: FTC Part 1

Find ddx[∫3π(t4−1)dt]\displaystyle\frac{d}{dx}\left[\int_3^\pi\left(t^4-1\right)dt\right]

The bounds are real numbers, not functions of xx. Therefore the integral will be evaluated to a number (a constant). But derivative of a constant is zero, so we have:

ddx∫3π(t4−1)dt=0\displaystyle\frac{d}{dx}\int_3^\pi\left(t^4-1\right)dt=0
Find the derivative ddx\displaystyle \frac{d}{dx} of the integral ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.

Compute f(π/3)f(\pi/3)given that f(x)=ddx∫sin⁡xcos⁡xln⁡t dt\displaystyle f(x)=\frac{d}{dx}\int_{\sin x}^{\cos x}\ln t\ dt .
Extra Practice
Practice: FTC I (~F2016 Final Q40)
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Derivative of Integral
Find ddx∫x3(t2−tan⁡t)100dt\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC I
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Application of Fundamental Theorem of Calculus
For what values of xx is the function f(x)=∫x22(t−1)dtf\left(x\right)=\int_{x^2}^2\left(t-1\right)dt decreasing?
Practice: Application of FTC I (~F2015 Final Q16)
Practice: Application of FTC I
Given that lim⁡x→0∫0x(t2−1)dt=0, \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}.
Practice: FTC I (~F2016 Final Q40)
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Practice: FTC I
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
simona integral ftc
If g(x)=∫1xcos⁡t dt\displaystyle g\left(x\right)=\int_1^x\cos t \ dt, find g′(π2)g'\left(\dfrac{\pi}{2}\right).
Evaluate Derivative of Integral
Given f(x)=ddt∫sin⁡xcos⁡xln⁡t dt\displaystyle f(x)=\frac{d}{dt}\int_{\sin x}^{\cos x}\ln t\ dt , find f(π/3)f(\pi/3).
Practice: Deriv of Integral with Chain Rule
Q.\textbf{Q.} Find the derivative of ∫0ln⁡(x2)sin⁡t2 dt\displaystyle \int_{0}^{\ln(x^2)}\sin{t^2}\,\text{d}t.
Derivative of Integral
Find ddx∫x3(t2−tan⁡t)100dt\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
Practice: Application of FTC I (~F2018 Final Q40)
Practice: Application of FTC I
For what values of xx is the function f(x)=∫x22(t−1)dtf\left(x\right)=\int_{x^2}^2\left(t-1\right)dt decreasing?
Intervals of Increase and Decrease
If the function is given by f(x)=∫x2(t+1)2(t−2)t dtf(x)=\int_x^2\left(t+1\right)^2\left(t-2\right)t\ dt, then f(x)f\left(x\right) is decreasing on the interval
Applications of Integration: Position, Velocity and Acceleration
If the displacement of a particle is given by s(t)=∫0tx2sin⁡xdxs\left(t\right)=\int_0^t\frac{x^2}{\sin x}dx, then the acceleration at at t=π2t=\frac{\pi}{2} is
Find ddx(∫x3x20 dt)\displaystyle\frac{d}{dx}\left(\int_x^{3x^2}0\ dt\right).
Evaluate lim⁡x→0∫0xcos⁡2t dt2x\displaystyle\lim_{x\rightarrow0}\frac{\int_0^x\cos^2t\ dt}{2x} if it is known that lim⁡x→0∫0xcos⁡2t dt=0\displaystyle\lim_{x\rightarrow0}\int_0^x\cos^2t\ dt=0.
Practice: FTC
Find the derivative of ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
Practice: FTC
Q.\textbf{Q.} Find the derivative of ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Find the derivative using the Fundamental Theorem of Calculus.
ddx∫x3csc⁡t dt\displaystyle\frac{d}{dx}\int_x^3\csc t\ dt
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Derivatives
Find the derivative of
∫−12x3cos⁡t2 dt\int_{-1}^{2x^3} \cos t^2\ dt
Derivatives
Find the derivative ddx∫0sin⁡πsin⁡t2 dt\displaystyle \dfrac{d}{dx}\int_{0}^{\sin\pi}\sin{t^2}\,\text{d}t.
Derivatives
Given that lim⁡x→0∫0x(t2−1)dt=0, \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}.
Practice: FTC I
If g(x)=∫x2excos⁡t dt\displaystyle g\left(x\right)=\int_{x^2}^{e^x}\cos t\ dt, find g′(0)g'\left(0\right).
Practice: Application of FTC I
Given that lim⁡x→0∫0x(t2−1)dt=0, \displaystyle \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\displaystyle \lim_{x\rightarrow0}\frac{\displaystyle \int_0^x\left(t^2-1\right)dt}{\sin x}.
Find the derivative using the Fundamental Theorem of Calculus: ddx∫sin⁡xcos⁡xln⁡t dt\displaystyle\frac{d}{dx}\int_{\sin x}^{\cos x}\ln t\ dt.
Find the derivative using the Fundamental Theorem of Calculus: ddx∫x3csc⁡t dt\displaystyle\frac{d}{dx}\int_x^3\csc t\ dt.
Compute the derivative ddx[∫x21(et+1)dt]\displaystyle \frac{d}{dx}\left[\int_{x^2}^{1}(e^t+1)dt\right].
Trigonometric Functions
Let f(x)=1+∫55xarctan⁡(t2)dtf\left(x\right)=1+\int_5^{5x}\arctan\left(t^2\right)dt
Find the linearization of f(x)f\left(x\right) centered at a=1a=1.
simona integral ftc
If g(x)=∫1xcos⁡t dt\displaystyle g\left(x\right)=\int_1^x\cos t \ dt, find g′(π2)g'\left(\dfrac{\pi}{2}\right).
Practice: Deriv of Integral with Chain Rule
Q.\textbf{Q.} Find the derivative of ∫0ln⁡(x2)sin⁡t2 dt\displaystyle \int_{0}^{\ln(x^2)}\sin{t^2}\,\text{d}t.
Evaluate Derivative of Integral
Given f(x)=ddt∫sin⁡xcos⁡xln⁡t dt\displaystyle f(x)=\frac{d}{dt}\int_{\sin x}^{\cos x}\ln t\ dt , find f(π/3)f(\pi/3).
Derivative of Integral
Find ddx∫x3(t2−tan⁡t)100dt\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
Find the derivative using the Fundamental Theorem of Calculus: ddx∫x3xarctan⁡ t dt\displaystyle\frac{d}{dx}\int_{\sqrt{x}}^{3x}\arctan\ t\ dt.
Find the derivative using the Fundamental Theorem of Calculus: ddx∫3x9xlog⁡3t dt\displaystyle\frac{d}{dx}\int_{3^x}^{9^x}\log_3t\ dt.
Find the second derivative with respect to x of ∫46xt1+tdt\int_4^{6x}\frac{t}{1+t}dt
Find the derivative of the integral, i.e.
ddx∫sin⁡xcos⁡xln⁡t dt\displaystyle\frac{d}{dx}\int_{\sin x}^{\cos x}\ln t\ dt
Find the derivative with respect to xx of ∫π/6ln⁡x(sec⁡t+tan⁡t)dt\int_{\pi/6}^{\ln x}(\sec t+\tan t)dt.
Find the second derivative with respect to xx of ∫46xt1+tdt\int_4^{6x}\frac{t}{1+t}dt.
Find ddx(∫x3x20 dt)\displaystyle\frac{d}{dx}\left(\int_x^{3x^2}0\ dt\right).
Evaluate lim⁡x→0∫0xcos⁡2t dt2x\displaystyle\lim_{x\rightarrow0}\frac{\int_0^x\cos^2t\ dt}{2x} if it is known that lim⁡x→0∫0xcos⁡2t dt=0\displaystyle\lim_{x\rightarrow0}\int_0^x\cos^2t\ dt=0.
Applications of Integration: Position, Velocity and Acceleration
A particle's velocity in cm/s is given by the function v(t)=∫1t2x dxv\left(t\right)=\int_1^{t^2}x\ dx. When is the particle's acceleration 5454cm/s2?
Derivatives
Find ddx[∫x3x(t−t3) dt]\frac{d}{dx}\left[\int_{x^3}^{\sqrt{x}}\left(\sqrt{t}-t^3\right)\ dt\right].
Application of Fundamental Theorem of Calculus
For what values of xx is the function f(x)=∫x22(t−1)dtf\left(x\right)=\int_{x^2}^2\left(t-1\right)dt decreasing?
Practice: Application of FTC I (~F2015 Final Q16)
Practice: Application of FTC I
Given that lim⁡x→0∫0x(t2−1)dt=0, \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\ determine lim⁡x→0∫0x(t2−1)dtsin⁡x\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}.
Practice: FTC I (~F2016 Final Q40)
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC I
Find ddx(∫xx2t2dt)\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right).
Practice: FTC I
If g(x)=∫1xcos⁡t dtg\left(x\right)=\int_1^x\cos t\ dt, find g′(π2)g'\left(\frac{\pi}{2}\right).
Practice: FTC
Q.\textbf{Q.} Find the derivative of ∫0x2sin⁡t2 dt\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t.