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# Integral Table 2/2

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• Integrator

### Basic Forms

 $\int {x}^{n}dx=\frac{1}{n+1}{x}^{n+1},\phantom{\rule{5.16667pt}{0ex}}n\ne -1$ (1)
 $\int \frac{1}{x}dx=ln|x|$ (2)
 $\int udv=uv-\int vdu$ (3)
 $\int \frac{1}{ax+b}dx=\frac{1}{a}ln|ax+b|$ (4)

### Integrals of Rational Functions

 $\int \frac{1}{{\left(x+a\right)}^{2}}dx=-\frac{1}{x+a}$ (5)
 $\int {\left(x+a\right)}^{n}dx=\frac{{\left(x+a\right)}^{n+1}}{n+1},n\ne -1$ (6)
 $\int x{\left(x+a\right)}^{n}dx=\frac{{\left(x+a\right)}^{n+1}\left(\left(n+1\right)x-a\right)}{\left(n+1\right)\left(n+2\right)}$ (7)
 $\int \frac{1}{1+{x}^{2}}dx={tan}^{-1}x$ (8)
 $\int \frac{1}{{a}^{2}+{x}^{2}}dx=\frac{1}{a}{tan}^{-1}\frac{x}{a}$ (9)
 $\int \frac{x}{{a}^{2}+{x}^{2}}dx=\frac{1}{2}ln|{a}^{2}+{x}^{2}|$ (10)
 $\int \frac{{x}^{2}}{{a}^{2}+{x}^{2}}dx=x-a{tan}^{-1}\frac{x}{a}$ (11)
 $\int \frac{{x}^{3}}{{a}^{2}+{x}^{2}}dx=\frac{1}{2}{x}^{2}-\frac{1}{2}{a}^{2}ln|{a}^{2}+{x}^{2}|$ (12)
 $\int \frac{1}{a{x}^{2}+bx+c}dx=\frac{2}{\sqrt{4ac-{b}^{2}}}{tan}^{-1}\frac{2ax+b}{\sqrt{4ac-{b}^{2}}}$ (13)
 (14)
 $\int \frac{x}{{\left(x+a\right)}^{2}}dx=\frac{a}{a+x}+ln|a+x|$ (15)
 $\int \frac{x}{a{x}^{2}+bx+c}dx=\frac{1}{2a}ln|a{x}^{2}+bx+c|-\frac{b}{a\sqrt{4ac-{b}^{2}}}{tan}^{-1}\frac{2ax+b}{\sqrt{4ac-{b}^{2}}}$ (16)

### Integrals with Roots

 $\int \sqrt{x-a}\phantom{\rule{1em}{0ex}}dx=\frac{2}{3}{\left(x-a\right)}^{3∕2}$ (17)
 $\int \frac{1}{\sqrt{x±a}}\phantom{\rule{1em}{0ex}}dx=2\sqrt{x±a}$ (18)
 $\int \frac{1}{\sqrt{a-x}}\phantom{\rule{1em}{0ex}}dx=-2\sqrt{a-x}$ (19)
 (20)
 $\int \sqrt{ax+b}\phantom{\rule{1em}{0ex}}dx=\left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b}$ (21)
 $\int {\left(ax+b\right)}^{3∕2}\phantom{\rule{1em}{0ex}}dx=\frac{2}{5a}{\left(ax+b\right)}^{5∕2}$ (22)
 $\int \frac{x}{\sqrt{x±a}}\phantom{\rule{1em}{0ex}}dx=\frac{2}{3}\left(x\mp 2a\right)\sqrt{x±a}$ (23)
 $\int \sqrt{\frac{x}{a-x}}\phantom{\rule{1em}{0ex}}dx=-\sqrt{x\left(a-x\right)}-a{tan}^{-1}\frac{\sqrt{x\left(a-x\right)}}{x-a}$ (24)
 $\int \sqrt{\frac{x}{a+x}}\phantom{\rule{1em}{0ex}}dx=\sqrt{x\left(a+x\right)}-aln\left[\sqrt{x}+\sqrt{x+a}\right]$ (25)
 $\int x\sqrt{ax+b}\phantom{\rule{1em}{0ex}}dx=\frac{2}{15{a}^{2}}\left(-2{b}^{2}+abx+3{a}^{2}{x}^{2}\right)\sqrt{ax+b}$ (26)
 $\int \sqrt{x\left(ax+b\right)}\phantom{\rule{1em}{0ex}}dx=\frac{1}{4{a}^{3∕2}}\left[\left(2ax+b\right)\sqrt{ax\left(ax+b\right)}-{b}^{2}ln\left|a\sqrt{x}+\sqrt{a\left(ax+b\right)}\right|\right]$ (27)
 $\int \sqrt{{x}^{3}\left(ax+b\right)}\phantom{\rule{1em}{0ex}}dx=\left[\frac{b}{12a}-\frac{{b}^{2}}{8{a}^{2}x}+\frac{x}{3}\right]\sqrt{{x}^{3}\left(ax+b\right)}+\frac{{b}^{3}}{8{a}^{5∕2}}ln\left|a\sqrt{x}+\sqrt{a\left(ax+b\right)}\right|$ (28)
 $\int \sqrt{{x}^{2}±{a}^{2}}\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}x\sqrt{{x}^{2}±{a}^{2}}±\frac{1}{2}{a}^{2}ln\left|x+\sqrt{{x}^{2}±{a}^{2}}\right|$ (29)
 $\int \sqrt{{a}^{2}-{x}^{2}}\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}x\sqrt{{a}^{2}-{x}^{2}}+\frac{1}{2}{a}^{2}{tan}^{-1}\frac{x}{\sqrt{{a}^{2}-{x}^{2}}}$ (30)
 $\int x\sqrt{{x}^{2}±{a}^{2}}\phantom{\rule{1em}{0ex}}dx=\frac{1}{3}{\left({x}^{2}±{a}^{2}\right)}^{3∕2}$ (31)
 $\int \frac{1}{\sqrt{{x}^{2}±{a}^{2}}}\phantom{\rule{1em}{0ex}}dx=ln\left|x+\sqrt{{x}^{2}±{a}^{2}}\right|$ (32)
 $\int \frac{1}{\sqrt{{a}^{2}-{x}^{2}}}\phantom{\rule{1em}{0ex}}dx={sin}^{-1}\frac{x}{a}$ (33)
 $\int \frac{x}{\sqrt{{x}^{2}±{a}^{2}}}\phantom{\rule{1em}{0ex}}dx=\sqrt{{x}^{2}±{a}^{2}}$ (34)
 $\int \frac{x}{\sqrt{{a}^{2}-{x}^{2}}}\phantom{\rule{1em}{0ex}}dx=-\sqrt{{a}^{2}-{x}^{2}}$ (35)
 $\int \frac{{x}^{2}}{\sqrt{{x}^{2}±{a}^{2}}}\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}x\sqrt{{x}^{2}±{a}^{2}}\mp \frac{1}{2}{a}^{2}ln\left|x+\sqrt{{x}^{2}±{a}^{2}}\right|$ (36)
 $\int \sqrt{a{x}^{2}+bx+c}\phantom{\rule{1em}{0ex}}dx=\frac{b+2ax}{4a}\sqrt{a{x}^{2}+bx+c}+\frac{4ac-{b}^{2}}{8{a}^{3∕2}}ln\left|2ax+b+2\sqrt{a\left(a{x}^{2}+b{x}^{+}c\right)}\right|$ (37)
 $\begin{array}{cc}\begin{array}{rl}\int & x\sqrt{a{x}^{2}+bx+c}\phantom{\rule{1em}{0ex}}dx=\frac{1}{48{a}^{5∕2}}\left(2\sqrt{a}\sqrt{a{x}^{2}+bx+c}\right\left(-3{b}^{2}+2abx+8a\left(c+a{x}^{2}\right)\right)\\ & +3\left({b}^{3}-4abc\right)ln\left|b+2ax+2\sqrt{a}\sqrt{a{x}^{2}+bx+c}\right|)\end{array}& \end{array}$ (38)
 $\int \frac{1}{\sqrt{a{x}^{2}+bx+c}}\phantom{\rule{1em}{0ex}}dx=\frac{1}{\sqrt{a}}ln\left|2ax+b+2\sqrt{a\left(a{x}^{2}+bx+c\right)}\right|$ (39)
 $\int \frac{x}{\sqrt{a{x}^{2}+bx+c}}\phantom{\rule{1em}{0ex}}dx=\frac{1}{a}\sqrt{a{x}^{2}+bx+c}-\frac{b}{2{a}^{3∕2}}ln\left|2ax+b+2\sqrt{a\left(a{x}^{2}+bx+c\right)}\right|$ (40)
 $\int \frac{dx}{{\left({a}^{2}+{x}^{2}\right)}^{3∕2}}=\frac{x}{{a}^{2}\sqrt{{a}^{2}+{x}^{2}}}$ (41)

### Integrals with Logarithms

 $\int lnax\phantom{\rule{1em}{0ex}}dx=xlnax-x$ (42)
 $\int xlnx\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{x}^{2}lnx-\frac{{x}^{2}}{4}$ (43)
 $\int {x}^{2}lnx\phantom{\rule{1em}{0ex}}dx=\frac{1}{3}{x}^{3}lnx-\frac{{x}^{3}}{9}$ (44)
 $\int {x}^{n}lnx\phantom{\rule{1em}{0ex}}dx={x}^{n+1}\left(\frac{lnx}{n+1}-\frac{1}{{\left(n+1\right)}^{2}}\right),\phantom{\rule{10.33334pt}{0ex}}n\ne -1$ (45)
 $\int \frac{lnax}{x}\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{\left(lnax\right)}^{2}$ (46)
 $\int \frac{lnx}{{x}^{2}}\phantom{\rule{1em}{0ex}}dx=-\frac{1}{x}-\frac{lnx}{x}$ (47)
 $\int ln\left(ax+b\right)\phantom{\rule{1em}{0ex}}dx=\left(x+\frac{b}{a}\right)ln\left(ax+b\right)-x,a\ne 0$ (48)
 $\int ln\left({x}^{2}+{a}^{2}\right)\phantom{\rule{2.58333pt}{0ex}}dx=xln\left({x}^{2}+{a}^{2}\right)+2a{tan}^{-1}\frac{x}{a}-2x$ (49)
 $\int ln\left({x}^{2}-{a}^{2}\right)\phantom{\rule{2.58333pt}{0ex}}dx=xln\left({x}^{2}-{a}^{2}\right)+aln\frac{x+a}{x-a}-2x$ (50)
 $\int ln\left(a{x}^{2}+bx+c\right)\phantom{\rule{1em}{0ex}}dx=\frac{1}{a}\sqrt{4ac-{b}^{2}}{tan}^{-1}\frac{2ax+b}{\sqrt{4ac-{b}^{2}}}-2x+\left(\frac{b}{2a}+x\right)ln\left(a{x}^{2}+bx+c\right)$ (51)
 $\int xln\left(ax+b\right)\phantom{\rule{1em}{0ex}}dx=\frac{bx}{2a}-\frac{1}{4}{x}^{2}+\frac{1}{2}\left({x}^{2}-\frac{{b}^{2}}{{a}^{2}}\right)ln\left(ax+b\right)$ (52)
 $\int xln\left({a}^{2}-{b}^{2}{x}^{2}\right)\phantom{\rule{1em}{0ex}}dx=-\frac{1}{2}{x}^{2}+\frac{1}{2}\left({x}^{2}-\frac{{a}^{2}}{{b}^{2}}\right)ln\left({a}^{2}-{b}^{2}{x}^{2}\right)$ (53)
 $\int {\left(lnx\right)}^{2}\phantom{\rule{1em}{0ex}}dx=2x-2xlnx+x{\left(lnx\right)}^{2}$ (54)
 $\int {\left(lnx\right)}^{3}\phantom{\rule{1em}{0ex}}dx=-6x+x{\left(lnx\right)}^{3}-3x{\left(lnx\right)}^{2}+6xlnx$ (55)
 $\int x{\left(lnx\right)}^{2}\phantom{\rule{1em}{0ex}}dx=\frac{{x}^{2}}{4}+\frac{1}{2}{x}^{2}{\left(lnx\right)}^{2}-\frac{1}{2}{x}^{2}lnx$ (56)
 $\int {x}^{2}{\left(lnx\right)}^{2}\phantom{\rule{1em}{0ex}}dx=\frac{2{x}^{3}}{27}+\frac{1}{3}{x}^{3}{\left(lnx\right)}^{2}-\frac{2}{9}{x}^{3}lnx$ (57)

### Integrals with Exponentials

 $\int {e}^{ax}\phantom{\rule{1em}{0ex}}dx=\frac{1}{a}{e}^{ax}$ (58)
 (59)
 $\int x{e}^{x}\phantom{\rule{1em}{0ex}}dx=\left(x-1\right){e}^{x}$ (60)
 $\int x{e}^{ax}\phantom{\rule{1em}{0ex}}dx=\left(\frac{x}{a}-\frac{1}{{a}^{2}}\right){e}^{ax}$ (61)
 $\int {x}^{2}{e}^{x}\phantom{\rule{1em}{0ex}}dx=\left({x}^{2}-2x+2\right){e}^{x}$ (62)
 $\int {x}^{2}{e}^{ax}\phantom{\rule{1em}{0ex}}dx=\left(\frac{{x}^{2}}{a}-\frac{2x}{{a}^{2}}+\frac{2}{{a}^{3}}\right){e}^{ax}$ (63)
 $\int {x}^{3}{e}^{x}\phantom{\rule{1em}{0ex}}dx=\left({x}^{3}-3{x}^{2}+6x-6\right){e}^{x}$ (64)
 $\int {x}^{n}{e}^{ax}\phantom{\rule{1em}{0ex}}dx=\frac{{x}^{n}{e}^{ax}}{a}-\frac{n}{a}\int {x}^{n-1}{e}^{ax}\phantom{\rule{1.0pt}{0ex}}\text{d}x$ (65)
 (66)
 $\int {e}^{a{x}^{2}}\phantom{\rule{1em}{0ex}}dx=-\frac{i\sqrt{\pi }}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right)$ (67)
 $\int {e}^{-a{x}^{2}}\phantom{\rule{1em}{0ex}}dx=\frac{\sqrt{\pi }}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right)$ (68)
 $\int x{e}^{-a{x}^{2}}\phantom{\rule{1em}{0ex}}dx=-\frac{1}{2a}{e}^{-a{x}^{2}}$ (69)
 $\int {x}^{2}{e}^{-a{x}^{2}}\phantom{\rule{1em}{0ex}}dx=\frac{1}{4}\sqrt{\frac{\pi }{{a}^{3}}}\text{erf}\left(x\sqrt{a}\right)-\frac{x}{2a}{e}^{-a{x}^{2}}$ (70)

### Integrals with Trigonometric Functions

 $\int sinax\phantom{\rule{1em}{0ex}}dx=-\frac{1}{a}cosax$ (71)
 $\int {sin}^{2}ax\phantom{\rule{1em}{0ex}}dx=\frac{x}{2}-\frac{sin2ax}{4a}$ (72)
 $\int {sin}^{3}ax\phantom{\rule{1em}{0ex}}dx=-\frac{3cosax}{4a}+\frac{cos3ax}{12a}$ (73)
 $\int {sin}^{n}ax\phantom{\rule{1em}{0ex}}dx=-\frac{1}{a}cosax{\phantom{\rule{5.69054pt}{0ex}}}_{2}{F}_{1}\left[\frac{1}{2},\frac{1-n}{2},\frac{3}{2},{cos}^{2}ax\right]$ (74)
 $\int cosax\phantom{\rule{1em}{0ex}}dx=\frac{1}{a}sinax$ (75)
 $\int {cos}^{2}ax\phantom{\rule{1em}{0ex}}dx=\frac{x}{2}+\frac{sin2ax}{4a}$ (76)
 $\int {cos}^{3}axdx=\frac{3sinax}{4a}+\frac{sin3ax}{12a}$ (77)
 $\int {cos}^{p}axdx=-\frac{1}{a\left(1+p\right)}{cos}^{1+p}ax{×}_{2}{F}_{1}\left[\frac{1+p}{2},\frac{1}{2},\frac{3+p}{2},{cos}^{2}ax\right]$ (78)
 $\int cosxsinx\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{sin}^{2}x+{c}_{1}=-\frac{1}{2}{cos}^{2}x+{c}_{2}=-\frac{1}{4}cos2x+{c}_{3}$ (79)
 $\int cosaxsinbx\phantom{\rule{1em}{0ex}}dx=\frac{cos\left[\left(a-b\right)x\right]}{2\left(a-b\right)}-\frac{cos\left[\left(a+b\right)x\right]}{2\left(a+b\right)},a\ne b$ (80)
 $\int {sin}^{2}axcosbx\phantom{\rule{1em}{0ex}}dx=-\frac{sin\left[\left(2a-b\right)x\right]}{4\left(2a-b\right)}+\frac{sinbx}{2b}-\frac{sin\left[\left(2a+b\right)x\right]}{4\left(2a+b\right)}$ (81)
 $\int {sin}^{2}xcosx\phantom{\rule{1em}{0ex}}dx=\frac{1}{3}{sin}^{3}x$ (82)
 $\int {cos}^{2}axsinbx\phantom{\rule{1em}{0ex}}dx=\frac{cos\left[\left(2a-b\right)x\right]}{4\left(2a-b\right)}-\frac{cosbx}{2b}-\frac{cos\left[\left(2a+b\right)x\right]}{4\left(2a+b\right)}$ (83)
 $\int {cos}^{2}axsinax\phantom{\rule{1em}{0ex}}dx=-\frac{1}{3a}{cos}^{3}ax$ (84)
 $\int {sin}^{2}ax{cos}^{2}bxdx=\frac{x}{4}-\frac{sin2ax}{8a}-\frac{sin\left[2\left(a-b\right)x\right]}{16\left(a-b\right)}+\frac{sin2bx}{8b}-\frac{sin\left[2\left(a+b\right)x\right]}{16\left(a+b\right)}$ (85)
 $\int {sin}^{2}ax{cos}^{2}ax\phantom{\rule{1em}{0ex}}dx=\frac{x}{8}-\frac{sin4ax}{32a}$ (86)
 $\int tanax\phantom{\rule{1em}{0ex}}dx=-\frac{1}{a}lncosax$ (87)
 $\int {tan}^{2}ax\phantom{\rule{1em}{0ex}}dx=-x+\frac{1}{a}tanax$ (88)
 $\int {tan}^{n}ax\phantom{\rule{1em}{0ex}}dx=\frac{{tan}^{n+1}ax}{a\left(1+n\right)}{×}_{2}{F}_{1}\left(\frac{n+1}{2},1,\frac{n+3}{2},-{tan}^{2}ax\right)$ (89)
 $\int {tan}^{3}axdx=\frac{1}{a}lncosax+\frac{1}{2a}{sec}^{2}ax$ (90)
 $\int secx\phantom{\rule{1em}{0ex}}dx=ln|secx+tanx|=2{tanh}^{-1}\left(tan\frac{x}{2}\right)$ (91)
 $\int {sec}^{2}ax\phantom{\rule{1em}{0ex}}dx=\frac{1}{a}tanax$ (92)
 $\int {sec}^{3}x\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}secxtanx+\frac{1}{2}ln|secx+tanx|$ (93)
 $\int secxtanx\phantom{\rule{1em}{0ex}}dx=secx$ (94)
 $\int {sec}^{2}xtanx\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{sec}^{2}x$ (95)
 $\int {sec}^{n}xtanx\phantom{\rule{1em}{0ex}}dx=\frac{1}{n}{sec}^{n}x,n\ne 0$ (96)
 $\int cscx\phantom{\rule{1em}{0ex}}dx=ln\left|tan\frac{x}{2}\right|=ln|cscx-cotx|+C$ (97)
 $\int {csc}^{2}ax\phantom{\rule{1em}{0ex}}dx=-\frac{1}{a}cotax$ (98)
 $\int {csc}^{3}x\phantom{\rule{1em}{0ex}}dx=-\frac{1}{2}cotxcscx+\frac{1}{2}ln|cscx-cotx|$ (99)
 $\int {csc}^{n}xcotx\phantom{\rule{1em}{0ex}}dx=-\frac{1}{n}{csc}^{n}x,n\ne 0$ (100)
 $\int secxcscx\phantom{\rule{1em}{0ex}}dx=ln|tanx|$ (101)

### Products of Trigonometric Functions and Monomials

 $\int xcosx\phantom{\rule{1em}{0ex}}dx=cosx+xsinx$ (102)
 $\int xcosax\phantom{\rule{1em}{0ex}}dx=\frac{1}{{a}^{2}}cosax+\frac{x}{a}sinax$ (103)
 $\int {x}^{2}cosx\phantom{\rule{1em}{0ex}}dx=2xcosx+\left({x}^{2}-2\right)sinx$ (104)
 $\int {x}^{2}cosax\phantom{\rule{1em}{0ex}}dx=\frac{2xcosax}{{a}^{2}}+\frac{{a}^{2}{x}^{2}-2}{{a}^{3}}sinax$ (105)
 $\int {x}^{n}cosxdx=-\frac{1}{2}{\left(i\right)}^{n+1}\left[\Gamma \left(n+1,-ix\right)+{\left(-1\right)}^{n}\Gamma \left(n+1,ix\right)\right]$ (106)
 $\int {x}^{n}cosax\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{\left(ia\right)}^{1-n}\left[{\left(-1\right)}^{n}\Gamma \left(n+1,-iax\right)-\Gamma \left(n+1,ixa\right)\right]$ (107)
 $\int xsinx\phantom{\rule{1em}{0ex}}dx=-xcosx+sinx$ (108)
 $\int xsinax\phantom{\rule{1em}{0ex}}dx=-\frac{xcosax}{a}+\frac{sinax}{{a}^{2}}$ (109)
 $\int {x}^{2}sinx\phantom{\rule{1em}{0ex}}dx=\left(2-{x}^{2}\right)cosx+2xsinx$ (110)
 $\int {x}^{2}sinax\phantom{\rule{1em}{0ex}}dx=\frac{2-{a}^{2}{x}^{2}}{{a}^{3}}cosax+\frac{2xsinax}{{a}^{2}}$ (111)
 $\int {x}^{n}sinx\phantom{\rule{1em}{0ex}}dx=-\frac{1}{2}{\left(i\right)}^{n}\left[\Gamma \left(n+1,-ix\right)-{\left(-1\right)}^{n}\Gamma \left(n+1,-ix\right)\right]$ (112)
 $\int x{cos}^{2}x\phantom{\rule{1em}{0ex}}dx=\frac{{x}^{2}}{4}+\frac{1}{8}cos2x+\frac{1}{4}xsin2x$ (113)
 $\int x{sin}^{2}x\phantom{\rule{1em}{0ex}}dx=\frac{{x}^{2}}{4}-\frac{1}{8}cos2x-\frac{1}{4}xsin2x$ (114)
 $\int x{tan}^{2}x\phantom{\rule{1em}{0ex}}dx=-\frac{{x}^{2}}{2}+lncosx+xtanx$ (115)
 $\int x{sec}^{2}x\phantom{\rule{1em}{0ex}}dx=lncosx+xtanx$ (116)

### Products of Trigonometric Functions and Exponentials

 $\int {e}^{x}sinx\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{e}^{x}\left(sinx-cosx\right)$ (117)
 $\int {e}^{bx}sinax\phantom{\rule{1em}{0ex}}dx=\frac{1}{{a}^{2}+{b}^{2}}{e}^{bx}\left(bsinax-acosax\right)$ (118)
 $\int {e}^{x}cosx\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{e}^{x}\left(sinx+cosx\right)$ (119)
 $\int {e}^{bx}cosax\phantom{\rule{1em}{0ex}}dx=\frac{1}{{a}^{2}+{b}^{2}}{e}^{bx}\left(asinax+bcosax\right)$ (120)
 $\int x{e}^{x}sinx\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{e}^{x}\left(cosx-xcosx+xsinx\right)$ (121)
 $\int x{e}^{x}cosx\phantom{\rule{1em}{0ex}}dx=\frac{1}{2}{e}^{x}\left(xcosx-sinx+xsinx\right)$ (122)

### Integrals of Hyperbolic Functions

 $\int coshax\phantom{\rule{1em}{0ex}}dx=\frac{1}{a}sinhax$ (123)
 $\int {e}^{ax}coshbx\phantom{\rule{1em}{0ex}}dx=\left\{\begin{array}{cc}\frac{{e}^{ax}}{{a}^{2}-{b}^{2}}\left[acoshbx-bsinhbx\right]\phantom{\rule{1em}{0ex}}\hfill & a\ne b\hfill \\ \frac{{e}^{2ax}}{4a}+\frac{x}{2}\phantom{\rule{1em}{0ex}}\hfill & a=b\hfill \end{array}\right\$ (124)
 $\int sinhax\phantom{\rule{1em}{0ex}}dx=\frac{1}{a}coshax$ (125)
 $\int {e}^{ax}sinhbx\phantom{\rule{1em}{0ex}}dx=\left\{\begin{array}{cc}\frac{{e}^{ax}}{{a}^{2}-{b}^{2}}\left[-bcoshbx+asinhbx\right]\phantom{\rule{1em}{0ex}}\hfill & a\ne b\hfill \\ \frac{{e}^{2ax}}{4a}-\frac{x}{2}\phantom{\rule{1em}{0ex}}\hfill & a=b\hfill \end{array}\right\$ (126)
 $\int tanhax\phantom{\rule{1.5pt}{0ex}}dx=\frac{1}{a}lncoshax$ (127)
 $\int {e}^{ax}tanhbx\phantom{\rule{1em}{0ex}}dx=\left\{\begin{array}{cc}\frac{{e}^{\left(a+2b\right)x}}{\left(a+2b\right)}\left({\phantom{\rule{1em}{0ex}}}_{2}{F}_{1}\right)\left[1+\frac{a}{2b},1,2+\frac{a}{2b},-{e}^{2bx}\right]\phantom{\rule{1em}{0ex}}\hfill & \hfill \\ \phantom{\rule{28.45274pt}{0ex}}-\frac{{e}^{ax}}{a}\left({\phantom{\rule{1em}{0ex}}}_{2}{F}_{1}\right)\left[1,\frac{a}{2b},1+\frac{a}{2b},-{e}^{2bx}\right]\phantom{\rule{1em}{0ex}}\hfill & a\ne b\hfill \\ \frac{{e}^{ax}-2{tan}^{-1}\left[{e}^{ax}\right]}{a}\phantom{\rule{1em}{0ex}}\hfill & a=b\hfill \end{array}\right\$ (128)
 $\int cosaxcoshbx\phantom{\rule{1em}{0ex}}dx=\frac{1}{{a}^{2}+{b}^{2}}\left[asinaxcoshbx+bcosaxsinhbx\right]$ (129)
 $\int cosaxsinhbx\phantom{\rule{1em}{0ex}}dx=\frac{1}{{a}^{2}+{b}^{2}}\left[bcosaxcoshbx+asinaxsinhbx\right]$ (130)
 $\int sinaxcoshbx\phantom{\rule{1em}{0ex}}dx=\frac{1}{{a}^{2}+{b}^{2}}\left[-acosaxcoshbx+bsinaxsinhbx\right]$ (131)
 $\int sinaxsinhbx\phantom{\rule{1em}{0ex}}dx=\frac{1}{{a}^{2}+{b}^{2}}\left[bcoshbxsinax-acosaxsinhbx\right]$ (132)
 $\int sinhaxcoshaxdx=\frac{1}{4a}\left[-2ax+sinh2ax\right]$ (133)
 $\int sinhaxcoshbx\phantom{\rule{1em}{0ex}}dx=\frac{1}{{b}^{2}-{a}^{2}}\left[bcoshbxsinhax-acoshaxsinhbx\right]$ (134)

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