d/dxintegral of_x^x^2 sqrt(1 + t^2) dt

2xsqrt(1 + x^4) - sqrt(1 + x^2)

Second fundamental theorem of calculus worksheet

Differentiating undoes integrating, so the derivative of an integral from a fixed number up to x is just the integrand with x in it. When the limit is a function of x rather than x itself, the chain rule comes with it. The sheets also read g(x) from a graph of f, and find f(b) by adding the integral of f' to f(a).

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name The Second Fundamental Theorem Date Period Answer each question. Differentiate without evaluating the integral where you can. Differentiate an integral with a variable limit Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$ Find $\frac{d}{d x} \int_{2}^{x^{2}} \sqrt{t^{3} + 1} d t$ Find $\frac{d}{d x} \int_{4}^{3 x^{2}} (\sin t) d t$ Find $\frac{d}{d x} \int_{3}^{x^{2}} \cos ⁣ (t^{3}) d t$ Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (\cos t) d t$ Find $\frac{d}{d x} \int_{2}^{x^{2}} (t^{3} - t) d t$ Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 1}) d t$ Find $\frac{d}{d x} \int_{2}^{x^{2}} \ln ⁣ (t^{2} + 1) d t$ Analyze an accumulation function from a graph The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (- 2)$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . On what open intervals is $g$ increasing? The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g (4)$ The graph of $f$ is shown and $g (x) = \int_{5}^{x} f (t) d t$ . Find $g^{''} (0)$ Find $f(b)$ from $f(a)$ and $f'$ $f^{'} (x) = 3 x^{2} - 4 x + 2$ and $f (0) = 1$ . Find $f (3)$ $f^{'} (x) = 6 x^{2} - 3$ and $f (0) = 5$ . Find $f (3)$ $f^{'} (x) = 3 x^{2} + 2$ and $f (0) = 0$ . Find $f (1)$ $f^{'} (x) = - 3 x^{2} + 4 x + 2$ and $f (0) = - 4$ . Find $f (3)$ Differentiate with $x$ in both limits $\frac{d}{d x} \int_{x}^{x^{2}} \sqrt{1 + t^{2}} d t$ $\frac{d}{d x} \int_{2 x}^{x^{3}} \cos t d t$ $\frac{d}{d x} \int_{x}^{x^{2}} \sin t d t$ $\frac{d}{d x} \int_{x}^{x^{2}} t^{2} d t$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Second Fundamental Theorem

Date Period

Answer each question. Differentiate without evaluating the integral where you can.

Differentiate an integral with a variable limit
Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$
Find $\frac{d}{d x} \int_{2}^{x^{2}} \sqrt{t^{3} + 1} d t$
Find $\frac{d}{d x} \int_{4}^{3 x^{2}} (\sin t) d t$
Find $\frac{d}{d x} \int_{3}^{x^{2}} \cos ⁣ (t^{3}) d t$
Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (\cos t) d t$
Find $\frac{d}{d x} \int_{2}^{x^{2}} (t^{3} - t) d t$
Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 1}) d t$
Find $\frac{d}{d x} \int_{2}^{x^{2}} \ln ⁣ (t^{2} + 1) d t$

Analyze an accumulation function from a graph
The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (- 2)$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . On what open intervals is $g$ increasing?
The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g (4)$
The graph of $f$ is shown and $g (x) = \int_{5}^{x} f (t) d t$ . Find $g^{''} (0)$

Find $f(b)$ from $f(a)$ and $f'$
$f^{'} (x) = 3 x^{2} - 4 x + 2$ and $f (0) = 1$ . Find $f (3)$
$f^{'} (x) = 6 x^{2} - 3$ and $f (0) = 5$ . Find $f (3)$
$f^{'} (x) = 3 x^{2} + 2$ and $f (0) = 0$ . Find $f (1)$
$f^{'} (x) = - 3 x^{2} + 4 x + 2$ and $f (0) = - 4$ . Find $f (3)$

Differentiate with $x$ in both limits
$\frac{d}{d x} \int_{x}^{x^{2}} \sqrt{1 + t^{2}} d t$
$\frac{d}{d x} \int_{2 x}^{x^{3}} \cos t d t$
$\frac{d}{d x} \int_{x}^{x^{2}} \sin t d t$
$\frac{d}{d x} \int_{x}^{x^{2}} t^{2} d t$



Sigma Prep · sigmaprep.io/worksheets Name The Second Fundamental TheoremAnswer keyVersion 1 Date Period Answer each question. Differentiate without evaluating the integral where you can. Differentiate an integral with a variable limit Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$ $8 x ⁣ (64 x^{6} + 6)$ Find $\frac{d}{d x} \int_{2}^{x^{2}} \sqrt{t^{3} + 1} d t$ $2 x \sqrt{x^{6} + 1}$ Find $\frac{d}{d x} \int_{4}^{3 x^{2}} (\sin t) d t$ $6 x \sin ⁣ (3 x^{2})$ Find $\frac{d}{d x} \int_{3}^{x^{2}} \cos ⁣ (t^{3}) d t$ $2 x \cos ⁣ (x^{6})$ Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (\cos t) d t$ $6 x \cos ⁣ (3 x^{2})$ Find $\frac{d}{d x} \int_{2}^{x^{2}} (t^{3} - t) d t$ $2 x^{7} - 2 x^{3}$ Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 1}) d t$ $4 x \sqrt{4 x^{4} + 1}$ Find $\frac{d}{d x} \int_{2}^{x^{2}} \ln ⁣ (t^{2} + 1) d t$ $2 x \ln ⁣ (x^{4} + 1)$ Analyze an accumulation function from a graph The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (- 2)$ $\frac{1}{2}$ The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $(- 1, 5)$ The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g (4)$ $- \frac{3}{2}$ The graph of $f$ is shown and $g (x) = \int_{5}^{x} f (t) d t$ . Find $g^{''} (0)$ $1$ Find $f(b)$ from $f(a)$ and $f'$ $f^{'} (x) = 3 x^{2} - 4 x + 2$ and $f (0) = 1$ . Find $f (3)$ $16$ $f^{'} (x) = 6 x^{2} - 3$ and $f (0) = 5$ . Find $f (3)$ $50$ $f^{'} (x) = 3 x^{2} + 2$ and $f (0) = 0$ . Find $f (1)$ $3$ $f^{'} (x) = - 3 x^{2} + 4 x + 2$ and $f (0) = - 4$ . Find $f (3)$ $- 7$ Differentiate with $x$ in both limits $\frac{d}{d x} \int_{x}^{x^{2}} \sqrt{1 + t^{2}} d t$ $2 x \sqrt{1 + x^{4}} - \sqrt{1 + x^{2}}$ $\frac{d}{d x} \int_{2 x}^{x^{3}} \cos t d t$ $3 x^{2} \cos ⁣ (x^{3}) - 2 \cos 2 x$ $\frac{d}{d x} \int_{x}^{x^{2}} \sin t d t$ $2 x \sin ⁣ (x^{2}) - \sin x$ $\frac{d}{d x} \int_{x}^{x^{2}} t^{2} d t$ $2 x^{5} - x^{2}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Second Fundamental TheoremAnswer keyVersion 1

Date Period

Answer each question. Differentiate without evaluating the integral where you can.

Differentiate an integral with a variable limit
Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$ $8 x ⁣ (64 x^{6} + 6)$
Find $\frac{d}{d x} \int_{2}^{x^{2}} \sqrt{t^{3} + 1} d t$ $2 x \sqrt{x^{6} + 1}$
Find $\frac{d}{d x} \int_{4}^{3 x^{2}} (\sin t) d t$ $6 x \sin ⁣ (3 x^{2})$
Find $\frac{d}{d x} \int_{3}^{x^{2}} \cos ⁣ (t^{3}) d t$ $2 x \cos ⁣ (x^{6})$
Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (\cos t) d t$ $6 x \cos ⁣ (3 x^{2})$
Find $\frac{d}{d x} \int_{2}^{x^{2}} (t^{3} - t) d t$ $2 x^{7} - 2 x^{3}$
Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 1}) d t$ $4 x \sqrt{4 x^{4} + 1}$
Find $\frac{d}{d x} \int_{2}^{x^{2}} \ln ⁣ (t^{2} + 1) d t$ $2 x \ln ⁣ (x^{4} + 1)$

Analyze an accumulation function from a graph
The graph of $f$ is shown and $g (x) = \int_{- 1}^{x} f (t) d t$ . Find $g^{''} (- 2)$ $\frac{1}{2}$
The graph of $f$ is shown and $g (x) = \int_{3}^{x} f (t) d t$ . On what open intervals is $g$ increasing? $(- 1, 5)$
The graph of $f$ is shown and $g (x) = \int_{1}^{x} f (t) d t$ . Find $g (4)$ $- \frac{3}{2}$
The graph of $f$ is shown and $g (x) = \int_{5}^{x} f (t) d t$ . Find $g^{''} (0)$ $1$

Find $f(b)$ from $f(a)$ and $f'$
$f^{'} (x) = 3 x^{2} - 4 x + 2$ and $f (0) = 1$ . Find $f (3)$ $16$
$f^{'} (x) = 6 x^{2} - 3$ and $f (0) = 5$ . Find $f (3)$ $50$
$f^{'} (x) = 3 x^{2} + 2$ and $f (0) = 0$ . Find $f (1)$ $3$
$f^{'} (x) = - 3 x^{2} + 4 x + 2$ and $f (0) = - 4$ . Find $f (3)$ $- 7$

Differentiate with $x$ in both limits
$\frac{d}{d x} \int_{x}^{x^{2}} \sqrt{1 + t^{2}} d t$ $2 x \sqrt{1 + x^{4}} - \sqrt{1 + x^{2}}$
$\frac{d}{d x} \int_{2 x}^{x^{3}} \cos t d t$ $3 x^{2} \cos ⁣ (x^{3}) - 2 \cos 2 x$
$\frac{d}{d x} \int_{x}^{x^{2}} \sin t d t$ $2 x \sin ⁣ (x^{2}) - \sin x$
$\frac{d}{d x} \int_{x}^{x^{2}} t^{2} d t$ $2 x^{5} - x^{2}$

How to do these

Find $\frac{d}{d x} \int_{1}^{x^{2}} (t^{3} + 1) d t$

Put the upper limit in for t: x squared cubed plus 1, which is x to the sixth plus 1.
The upper limit is x squared, whose derivative is 2x.
Multiply them: 2x times x to the sixth plus 1.

The answer is $2 x ⁣ (x^{6} + 1)$ .

Where students go wrong

Working the integral out first. The whole point is that you never have to; and when the limit is a function of x, forgetting to multiply by its derivative loses the chain rule.

Also called second fundamental theorem, derivative of an integral, variable upper limit or accumulation function.

What you can put on this worksheet

The second fundamental theorem: Find $\frac{d}{d x} \int_{4}^{x} (\sin t) d t$ → $\sin x$
g(x) = ∫ f from a graph of f: The graph of $f$ is shown and $g (x) = \int_{6}^{x} f (t) d t$ . Find $g^{'} (2)$ — the graph of f on [-1, 8] → $0$
f(b) from f(a) and f': $f^{'} (x) = - 6 x^{2} + 2 x - 3$ and $f (1) = 4$ . Find $f (3)$ → $- 46$
x in both limits: $\frac{d}{d x} \int_{x}^{x^{2}} t^{2} d t$ → $2 x^{5} - x^{2}$
An upper limit of kx: Find $\frac{d}{d x} \int_{1}^{5 x} \sqrt{t^{2} + 1} d t$ → $5 \sqrt{25 x^{2} + 1}$

Questions about these worksheets

Yes. Every second fundamental theorem of calculus sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the second fundamental theorem of calculus questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: the second fundamental theorem, g(x) = ∫ f from a graph of f, f(b) from f(a) and f', x in both limits and an upper limit of kx. Tick as many as you want and set how many of each, or let it spread them evenly.

Working the integral out first. The whole point is that you never have to; and when the limit is a function of x, forgetting to multiply by its derivative loses the chain rule.

Second fundamental theorem of calculus worksheet

Select your difficulty

Differentiate an integral with a variable limit

Analyze an accumulation function from a graph

Find f(b)f(b)f(b) from f(a)f(a)f(a) and f′f'f′

Differentiate with xxx in both limits

Differentiate an integral with a variable limit

Analyze an accumulation function from a graph

Find f(b)f(b)f(b) from f(a)f(a)f(a) and f′f'f′

Differentiate with xxx in both limits

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these second fundamental theorem of calculus worksheets free?+

Do the second fundamental theorem of calculus worksheets come with an answer key?+

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What grade are these second fundamental theorem of calculus worksheets for?+

Can I get a different version of the same second fundamental theorem of calculus worksheet?+

Is this a free alternative to Kuta Software for second fundamental theorem of calculus?+

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What kinds of question are on a second fundamental theorem of calculus worksheet?+

What do students get wrong with second fundamental theorem of calculus?+

Find $f(b)$ from $f(a)$ and $f'$

Differentiate with $x$ in both limits

Find $f(b)$ from $f(a)$ and $f'$

Differentiate with $x$ in both limits