Definite Integrals and Indefinite Integrals. ∫ undu √a + bu = 2un√a + bu b (2n + 1) − 2na b (2n + 1) ∫un − 1du √a + bu Most of them are not found in Gradsteyn-Ryzhik. Type in any integral to get the solution, steps and graph This website uses cookies to ensure you get the best experience. This page has been accessed 12,649 times. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience. Integral Table. The product of two integrals can be expressed as a double integral: I2= Z∞ −∞. {\displaystyle {\begin{aligned}\int _{0}^{1}e^{x\cdot \ln a+(1-x)\cdot \ln b}\,dx&=\int _{0}^{1}\left({\frac {a}{b}}\right)^{x}\cdot b\,dx\\&=\int _{0}^{1}a^{x}\cdot b^{1-x}\,dx\\&={\frac {a-b}{\ln a-\ln b}}\qquad {\text{for }}a>0,\ b>0,\ a\neq b\end{aligned}}} ∫√a + bu u2 du = − √a + bu u + b 2∫ du u√a + bu 111. Substitute      and the Feynman-Hibbs Integral can be calculated with Mathematica: To see a nice cancellation of singularities at work plot the next expression around c = negative Integer: …this is a special case of the next integral below (m = -1 / 2). Property 1: p∫q f(a) da = p∫q f(t) dt. Free definite integral calculator - solve definite integrals with all the steps. $,$ \int_{a}^{b} f ( x ) g ( x ) d x = f ( c ) \int\limits_{a}^{b} g ( x ) d x, $,$ \text{where } c \text{ is a number between } a \text{ and } b \text{ as long as } f(x) \text{ is continous between } a \text{ and } b, \text{ and } g(x) \ge 0 $,$ \frac{d}{d \alpha} \int_{\Phi_1 ( \alpha )}^{\Phi_2 ( \alpha ) } F ( x , \alpha ) d x = \int_{\Phi_1 ( \alpha )}^{\Phi_2 ( \alpha ) } \frac{\partial F}{\partial \alpha} d x + F ( \Phi_2 , \alpha ) \frac{d \Phi_1}{d \alpha} - F ( \Phi_1 , \alpha ) \frac{d \Phi_2}{d \alpha} $,$ \int_{a}^{\infty} \frac {d x}{x^2 + a^2} = \frac{\pi}{2a} $,$ \int_{0}^{\infty} \frac{x^{p-1} d x}{1 + x} = \frac{\pi}{\sin p \pi} \qquad 0