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.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name The Second Fundamental Theorem Date Period Differentiate each. Do not evaluate the integral. Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$ Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\sin t) d t$ Find $\frac{d}{d x} \int_{4}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 3}) d t$ Find $\frac{d}{d x} \int_{2}^{3 x^{2}} (t^{2} + 3) d t$ Find $\frac{d}{d x} \int_{7}^{3 x^{2}} (\sqrt{t^{2} + 4}) d t$ Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (t^{4} + 3) d t$ Find $\frac{d}{d x} \int_{8}^{2 x^{2}} (t^{4} + 1) d t$ Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (t^{2} + 5) d t$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 4}) d t$ Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (\cos t) d t$ Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (\cos t) d t$ Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\cos t) d t$ Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\cos t) d t$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sin t) d t$ Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$ Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (\sin t) d t$ Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\sqrt{t^{2} + 2}) d t$ Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (t^{3} + 4) d t$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (t^{2} + 2) d t$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Second Fundamental Theorem

Date Period

Differentiate each. Do not evaluate the integral.

Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$
Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\sin t) d t$
Find $\frac{d}{d x} \int_{4}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 3}) d t$
Find $\frac{d}{d x} \int_{2}^{3 x^{2}} (t^{2} + 3) d t$
Find $\frac{d}{d x} \int_{7}^{3 x^{2}} (\sqrt{t^{2} + 4}) d t$
Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (t^{4} + 3) d t$
Find $\frac{d}{d x} \int_{8}^{2 x^{2}} (t^{4} + 1) d t$
Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (t^{2} + 5) d t$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 4}) d t$
Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (\cos t) d t$
Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (\cos t) d t$
Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\cos t) d t$
Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\cos t) d t$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sin t) d t$
Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$
Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (\sin t) d t$
Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\sqrt{t^{2} + 2}) d t$
Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (t^{3} + 4) d t$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (t^{2} + 2) d t$



Sigma Prep · sigmaprep.io/worksheets Name The Second Fundamental TheoremAnswer keyVersion 1 Date Period Differentiate each. Do not evaluate the integral. Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$ $8 x (64 x^{6} + 6)$ Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\sin t) d t$ $8 x \sin (4 x^{2})$ Find $\frac{d}{d x} \int_{4}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$ $4 x \sqrt{4 x^{4} + 3}$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 3}) d t$ $8 x \sqrt{16 x^{4} + 3}$ Find $\frac{d}{d x} \int_{2}^{3 x^{2}} (t^{2} + 3) d t$ $6 x (9 x^{4} + 3)$ Find $\frac{d}{d x} \int_{7}^{3 x^{2}} (\sqrt{t^{2} + 4}) d t$ $6 x \sqrt{9 x^{4} + 4}$ Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (t^{4} + 3) d t$ $8 x (256 x^{8} + 3)$ Find $\frac{d}{d x} \int_{8}^{2 x^{2}} (t^{4} + 1) d t$ $4 x (16 x^{8} + 1)$ Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (t^{2} + 5) d t$ $6 x (9 x^{4} + 5)$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 4}) d t$ $8 x \sqrt{16 x^{4} + 4}$ Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (\cos t) d t$ $4 x \cos (2 x^{2})$ Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (\cos t) d t$ $8 x \cos (4 x^{2})$ Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\cos t) d t$ $8 x \cos (4 x^{2})$ Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\cos t) d t$ $8 x \cos (4 x^{2})$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sin t) d t$ $8 x \sin (4 x^{2})$ Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$ $4 x \sqrt{4 x^{4} + 3}$ Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (\sin t) d t$ $8 x \sin (4 x^{2})$ Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\sqrt{t^{2} + 2}) d t$ $8 x \sqrt{16 x^{4} + 2}$ Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (t^{3} + 4) d t$ $4 x (8 x^{6} + 4)$ Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (t^{2} + 2) d t$ $8 x (16 x^{4} + 2)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Second Fundamental TheoremAnswer keyVersion 1

Date Period

Differentiate each. Do not evaluate the integral.

Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (t^{3} + 6) d t$ $8 x (64 x^{6} + 6)$
Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\sin t) d t$ $8 x \sin (4 x^{2})$
Find $\frac{d}{d x} \int_{4}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$ $4 x \sqrt{4 x^{4} + 3}$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 3}) d t$ $8 x \sqrt{16 x^{4} + 3}$
Find $\frac{d}{d x} \int_{2}^{3 x^{2}} (t^{2} + 3) d t$ $6 x (9 x^{4} + 3)$
Find $\frac{d}{d x} \int_{7}^{3 x^{2}} (\sqrt{t^{2} + 4}) d t$ $6 x \sqrt{9 x^{4} + 4}$
Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (t^{4} + 3) d t$ $8 x (256 x^{8} + 3)$
Find $\frac{d}{d x} \int_{8}^{2 x^{2}} (t^{4} + 1) d t$ $4 x (16 x^{8} + 1)$
Find $\frac{d}{d x} \int_{3}^{3 x^{2}} (t^{2} + 5) d t$ $6 x (9 x^{4} + 5)$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sqrt{t^{2} + 4}) d t$ $8 x \sqrt{16 x^{4} + 4}$
Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (\cos t) d t$ $4 x \cos (2 x^{2})$
Find $\frac{d}{d x} \int_{6}^{4 x^{2}} (\cos t) d t$ $8 x \cos (4 x^{2})$
Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\cos t) d t$ $8 x \cos (4 x^{2})$
Find $\frac{d}{d x} \int_{2}^{4 x^{2}} (\cos t) d t$ $8 x \cos (4 x^{2})$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (\sin t) d t$ $8 x \sin (4 x^{2})$
Find $\frac{d}{d x} \int_{1}^{2 x^{2}} (\sqrt{t^{2} + 3}) d t$ $4 x \sqrt{4 x^{4} + 3}$
Find $\frac{d}{d x} \int_{5}^{4 x^{2}} (\sin t) d t$ $8 x \sin (4 x^{2})$
Find $\frac{d}{d x} \int_{7}^{4 x^{2}} (\sqrt{t^{2} + 2}) d t$ $8 x \sqrt{16 x^{4} + 2}$
Find $\frac{d}{d x} \int_{2}^{2 x^{2}} (t^{3} + 4) d t$ $4 x (8 x^{6} + 4)$
Find $\frac{d}{d x} \int_{1}^{4 x^{2}} (t^{2} + 2) d t$ $8 x (16 x^{4} + 2)$

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.

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.

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

The Second Fundamental Theorem

The Second Fundamental TheoremAnswer keyVersion 1

How to do these

Where students go wrong

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?+

Are these second fundamental theorem of calculus worksheets printable?+

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?+

Can I choose how hard the second fundamental theorem of calculus questions are?+

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