This is a list with the formulas and proofs in GR.

The table of integrals by Gradshteyn and Ryzhik is one of the most widely used ones.
It is our goal to present proofs of these formulas and put them in some context.

The whole collection of proofs will appear published in SCIENTIA, Series A.

If you want to help, please send the details to vhm at math.tulane.edu

If you experience a problem downloading a formula/proof email me at vhm at math.tulane.edu


Directory of notes on the integrals in GR.

Index of the papers with formulas in GR.

Thanks to those who contribute with a proof:

M. Albano, T. Amdeberhan, E. Beyerstedt, S. Boettner,
K. Boyadzhiev, L. Glasser, X. Guan, L. Jiu,
M. Knight, K. Kohl, C. Koutschan, A. MacLeod,
L. Medina, R. Posey, F. Stan, A. Straub

The notation not yet means that either I have not written the proof or I do not know how to do it.


2.3: The exponential function

Section 2.31 Forms containing $e^{ax}$

Section 2.311


Section 2.312

Section 2.313

Section 2.314


Section 2.315




3.1-3.2: Power and algebraic functions

Section 3.11 Rational functions

Section 3.112 contains 6 formulas.


3.12: Products of rational functions and expresssions that can be reduced to square roots of first and second-degree polynomials

Section 3.121 contains 3 formulas.


3.13-3.17: Expresssions that can be reduced to square roots of third and fourth-degree polynomials
and their products with rational functions.

Section 3.131 contains 8 formulas.


Section 3.132 contains 6 formulas.


Section 3.133 contains 18 formulas.


Section 3.134 contains 18 formulas.


Section 3.135 contains 12 formulas.

Section 3.136 contains 2 formulas.


Section 3.137 contains 8 formulas.

Section 3.138 contains 8 formulas.


Section 3.139 contains 28 formulas.


Section 3.141 contains 36 formulas.


Section 3.142 contains 36 formulas.


Section 3.143 contains 2 formulas.


Section 3.144 contains 12 formulas.


Section 3.145 contains 4 formulas.


Section 3.146 contains 3 formulas.


Section 3.147 contains 8 formulas.


Section 3.148 contains 8 formulas.


Section 3.149 contains 8 formulas.


Section 3.151 contains 8 formulas.


Section 3.152 contains 12 formulas.


Section 3.153 contains 10 formulas.


Section 3.154 contains 9 formulas.


Section 3.155 contains 9 formulas.


Section 3.156 contains 9 formulas.


Section 3.157 contains 12 formulas.


Section 3.158 contains 18 formulas.


Section 3.159 contains 18 formulas.


Section 3.161 contains 9 formulas.


Section 3.162 contains 18 formulas.



Section 3.163 contains 6 formulas.


Section 3.164 contains 8 formulas.


Section 3.165 contains 12 formulas.



Section 3.166 contains 23 formulas.


Section 3.167 contains 32 formulas.



Section 3.168 contains 72 formulas.


Section 3.169 contains 18 formulas.


Section 3.171 contains 18 formulas.


Section 3.172 contains 18 formulas.



Section 3.173 contains 2 formulas.


Section 3.174 contains 4 formulas.


Section 3.175 contains 2 formulas.


Section 3.181 contains 2 formulas.


Section 3.182 contains 2 formulas.


Section 3.183 contains 3 formulas.



Section 3.184 contains 2 formulas.


Section 3.185 contains 7 formulas.


Section 3.186 contains 5 formulas.



3.19-3.23: Combinations of powers of x and powers of binomials of the form a+bx

Section 3.191 contains 3 formulas.


Section 3.192 contains 4 formulas.

Section 3.193 contains a single formula.


Section 3.194 contains 8 formulas.

Section 3.195 contains a single formula.

Section 3.196 contains 5 formulas.

Section 3.197 contains 12 formulas.

Section 3.198 contains a single formula.

Section 3.199 contains a single formula.

Section 3.211 contains a single formula:

Section 3.212 contains a single formula:

Section 3.213 contains a single formula:

Section 3.214 contains a single formula:

Section 3.215 contains a single formula:

Section 3.216 contains 2 formulas.

Section 3.217 contains a single formula.

Section 3.218 contains a single formula.

Section 3.219 contains a single formula.

Section 3.221 contains 2 formulas.

Section 3.222 contains 2 formulas.

Section 3.223 contains 3 formulas:

Section 3.224 contains a single formula.

Section 3.225 contains 3 formulas.

Section 3.226 contains 2 formulas.

Section 3.227 contains 2 formulas:


Section 3.228 contains 6 formulas:

Section 3.229 contains a single formula.

Section 3.231 contains 6 formulas.


Section 3.232 contains a single formula.


Section 3.233 contains a single formula.


Section 3.234 contains 2 formulas.


Section 3.235 contains a single formula.


Section 3.236 contains a single formula:

Section 3.237 contains a single formula:

Section 3.238 contains 3 formulas:


3.24-3.27: Powers of x, binomials of the form .... and of polynomials in x

Section 3.241 contains 6 formulas.

Section 3.242 contains two formulas:

Section 3.243 contains a single formula:

Section 3.244 contains 4 formulas.

Section 3.245 contains a single formula.

Section 3.246 contains a single formula.

Section 3.247 contains 2 formulas.

Section 3.248 contains 6 formulas:

Section 3.249 contains 8 formulas:

Section 3.251 contains 11 formulas.

Section 3.252 contains 14 formulas:


Section 3.253 contains a single formula:

Section 3.254 contains 2 formulas:

Section 3.255 contains a single formula:

Section 3.256 contains 2 formulas:

Section 3.257 contains a single formula:

Section 3.258 contains 7 formulas:


Section 3.259 contains 3 formulas:


Section 3.261 contains 4 formulas:

Section 3.262 contains a single formula:

Section 3.263 contains a single formula:


Section 3.264 contains 7 formulas:

Section 3.265 contains a single formula.


Section 3.266 contains a single formula:


Section 3.267 contains 3 formulas.


Section 3.268 contains 3 formulas:


Section 3.269 contains 3 formulas.


Section 3.271 contains 5 formulas:


Section 3.272 contains 2 formulas.


Section 3.273 contains 4 formulas:


Section 3.274 contains 3 formulas:

Section 3.275 contains 4 formulas:

Section 3.276 contains 2 formulas:

Section 3.277 contains 7 formulas:

Section 3.278 contains a single formula:


3.31: Exponential functions.

Section 3.310 contains a single formula.

Section 3.311 contains 13 formulas.

Section 3.312 contains 3 formulas.

Section 3.313 contains 2 formulas.

Section 3.314 contains a single formula.

Section 3.315 contains 2 formulas:

Section 3.316 contains a single formula.

Section 3.317 contains 2 formulas.

Section 3.318 contains 2 formulas:


3.32-3.34: Exponentials of more complicated arguments.

Section 3.321 contains 7 formulas:


Section 3.322 contains 3 formulas:


Section 3.323 contains 3 formulas:


Section 3.324 contains 2 formulas.


Section 3.325 contains a single formula.


Section 3.326 contains 5 formulas:



Exponentials of exponentials.

Section 3.327 contains a single formula.


Section 3.328 contains a single formula.


Section 3.329 contains a single formula.


Section 3.331 contains 4 formulas.


Section 3.332 contains a single formula:


Section 3.333 contains 3 formulas:


Section 3.334 contains a single formula.


Exponentials of hyperbolic functions.

Section 3.335 contains a single formula:


Section 3.336 contains 3 formulas:


Section 3.337 contains 3 formulas:



Exponentials of trigonometric functions and logarithms.

Section 3.338 contains 5 formulas:


Section 3.339 contains a single formula.


Section 3.341 contains a single formula:

Section 3.342 contains a single formula.


3.35: Combinations of exponentials and rational functions.

Section 3.351 contains 9 formulas.

Section 3.352 contains 7 formulas:



Section 3.353 contains 5 formulas:


Section 3.354 contains 5 formulas:


Section 3.355 contains 4 formulas:


Section 3.356 contains 4 formulas:


Section 3.357 contains 6 formulas:


Section 3.358 contains 8 formulas:


Section 3.359 contains a single formula:



3.36-3.37: Combinations of exponentials and algebraic functions.

Section 3.361 contains 3 formulas.


Section 3.362 contains 2 formulas.


Section 3.363 contains 2 formulas.


Section 3.364 contains 3 formulas.


Section 3.365 contains 2 formulas:


Section 3.366 contains 3 formulas:


Section 3.367 contains a single formula:


Section 3.368 contains a single formula:


Section 3.369 contains a single formula.


Section 3.371 contains a single formula.


Section 3.372 contains a single formula.


Section 3.373 contains a single formula:


Section 3.374 contains two formulas:



3.38-3.39: Combinations of exponentials and arbitrary powers.

Section 3.381 contains 11 formulas:


Section 3.382 contains 7 formulas:


Section 3.383 contains 10 formulas:


Section 3.384 contains 9 formulas:


Section 3.385 contains a single formula:


Section 3.386 contains two formulas:


Section 3.387 contains 7 formulas:


Section 3.388 contains 4 formulas:


Section 3.389 contains 9 formulas:


Section 3.391 contains a single formula:


Section 3.392 contains 4 formulas:


Section 3.393 contains a single formula:


Section 3.394 contains a single formula:


Section 3.395 contains 3 formulas:



3.41-3.44: Combinations of rational functions of powers and exponentials.

Section 3.411 contains 32 formulas.


Section 3.412 contains a single formula:


Section 3.413 contains 3 formulas:


Section 3.414 contains a single formula:


Section 3.415 contains 4 formulas:


Section 3.416 contains 3 formulas:


Section 3.417 contains 2 formulas.


Section 3.418 contains 3 formulas:


Section 3.419 contains 7 formulas:


Section 3.421 contains 5 formulas:


Section 3.422 contains a single formula:


Section 3.423 contains 6 formulas:



Section 3.424 contains 5 formulas:



Section 3.425 contains 2 formulas:



Section 3.426 contains 2 formulas:


Section 3.427 contains 11 formulas:


Section 3.428 contains 7 formulas:



Section 3.429 contains a single formula.


Section 3.431 contains 2 formulas:


Section 3.432 contains 2 formulas:


Section 3.433 contains a single formula.


Section 3.434 contains 2 formulas.


Section 3.435 contains 4 formulas.


Section 3.436 contains a single formula.


Section 3.437 contains a single formula:


Section 3.438 contains 4 formulas:


Section 3.439 contains a single formula:


Section 3.441 contains a single formula:


Section 3.442 contains 3 formulas:


Section 3.443 contains 3 formulas:




3.45: Combinations of powers and algebraic functions of exponentials.

Section 3.451 contains 2 formulas.

Section 3.452 contains 5 formulas.

Section 3.453 contains 2 formulas:

Section 3.454 contains 2 formulas:

Section 3.455 contains 2 formulas:

Section 3.456 contains 2 formulas:

Section 3.457 contains 3 formulas.

Section 3.458 contains 2 formulas:


3.46-3.48: Combinations of exponentials of more complicated arguments and powers.

Section 3.461 contains 9 formulas:

Section 3.462 contains 25 formulas:


Section 3.463 contains a single formula.


Section 3.464 contains a single formula.


Section 3.465 contains a single formula.


Section 3.466 contains 3 formulas.


Section 3.467 contains a single formula.


Section 3.468 contains 2 formulas.


Section 3.469 contains 3 formulas:


Section 3.471 contains 16 formulas:


Section 3.472 contains 5 formulas.


Section 3.473 contains a single formula.


Section 3.474 contains 2 formulas:



Section 3.475 contains 3 formulas.


Section 3.476 contains 2 formulas.


Section 3.477 contains 2 formulas:


Section 3.478 contains 4 formulas:


Section 3.479 contains 2 formulas:


Section 3.481 contains 2 formulas.


Section 3.482 contains 3 formulas:


Section 3.483 contains a single formula:


Section 3.484 contains a single formula.


Section 3.485 contains a single formula:


Section 3.486 contains a single formula:


Section 3.487 contains a single formula:


Section 3.511 contains 10 formulas.


Section 3.512 contains 2 formulas.


Section 3.513 contains 4 formulas:


Section 3.514 contains 4 formulas:


Section 3.515 contains a single formula.


Section 3.516 contains 3 formulas:


Section 3.517 contains 2 formulas:


Section 3.518 contains 6 formulas:


Section 3.519 contains a single formula:


Section 3.521 contains 4 formulas.


Section 3.522 contains 10 formulas.


Section 3.523 contains 12 formulas.


Section 3.524 contains 23 formulas.


Section 3.525 contains 8 formulas:


Section 3.526 contains 3 formulas:


Section 3.527 contains 16 formulas:


Section 3.528 contains 2 formulas:


Section 3.529 contains 3 formulas:


Section 3.531 contains 8 formulas:



Section 3.532 contains 2 formulas:



Section 3.533 contains 4 formulas:


Section 3.534 contains 2 formulas:


Section 3.535 contains a single formula:



Section 3.536 contains 4 formulas:



Section 3.541 contains 10 formulas:


Section 3.542 contains 2 formulas:



Section 3.543 contains 4 formulas:



Section 3.544 contains a single formula:


Section 3.545 contains 2 formulas:


Section 3.546 contains 4 formulas.


Section 3.547 contains 20 formulas:



Section 3.548 contains 2 formulas:



Section 3.549 contains 4 formulas:



Section 3.551 contains 10 formulas:


Section 3.552 contains 6 formulas.



Section 3.553 contains 2 formulas.



Section 3.554 contains 6 formulas:



Section 3.555 contains 2 formulas:



Section 3.556 contains 2 formulas:



Section 3.557 contains 7 formulas:


Section 3.558 contains 7 formulas:


Section 3.559 contains a single formula:


Section 3.561 contains a single formula:


Section 3.562 contains 6 formulas:


Section 3.611 contains 3 formulas:


Section 3.612 contains 7 formulas:


Section 3.613 contains 13 formulas: