(i.)
| ∫∞0 | xa−1 | dx = | π | , (1 > a > 0), |
| 1 + x | sin aπ |
(ii.)
| ∫∞0 | xa−1 − xb−1 | dx = π (cot aπ − cot bπ), (0 < a or b < 1), |
| 1 − x |
(iii.)
| ∫∞0 | xa−1 log x | dx = | π2 | , (a > 1), |
| x − 1 | sin2 aπ |
(iv.)
∫∞0 x2 · cos 2x · e−x2 dx = −1⁄4 e−1 √π,
(v.)
| ∫10 | 1 − x2 | dx | = log tan | π | , | |
| 1 + x4 | log x | 8 |