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61.
∫sin⁡mxsin⁡nxdx=−sin⁡(m+n)x2(m+n)+sin⁡(m−n)x2(m−n)+C,m2≠n2\int \sin mx \sin nx dx = -\dfrac{\sin(m+n)x}{2(m+n)}+\dfrac{\sin(m-n)x}{2(m-n)} + C, m^2 \ne n^2
Indefinite Integral
Trigonometry
62.
∫cos⁡mxcos⁡nxdx=sin⁡(m+n)x2(m+n)+sin⁡(m−n)x2(m−n)+C,m2≠n2\int \cos mx \cos nx dx = \dfrac{\sin(m+n)x}{2(m+n)}+\dfrac{\sin(m-n)x}{2(m-n)} + C, m^2 \ne n^2
Indefinite Integral
Trigonometry
63.
∫sin⁡xcos⁡nx=−cos⁡n+1xn+1+C\int \sin x \cos^nx= - \dfrac{\cos^{n+1}x}{n+1}+C
Indefinite Integral
Trigonometry
64.
∫sin⁡nxcos⁡x=sin⁡n+1xn+1+C\int \sin^n x \cos x= \dfrac{\sin^{n+1}x}{n+1}+C
Indefinite Integral
Trigonometry
65.
Integral of arcsin
∫arcsin⁡xdx=xarcsin⁡x+1−x2+C\int \arcsin x dx = x\arcsin x + \sqrt{1-x^2}+C
Indefinite Integral
Trigonometry
66.
∫sin⁡mxcos⁡nxdx=−cos⁡(m+n)x2(m+n)−cos⁡(m−n)x2(m−n)+C,m2≠n2\int \sin mx \cos nx dx = -\dfrac{\cos(m+n)x}{2(m+n)}-\dfrac{\cos(m-n)x}{2(m-n)} + C, m^2 \ne n^2
Indefinite Integral
Trigonometry
67.
Integral of arccos
∫arccos⁡xdx=xarccos⁡x−1−x2+C\int \arccos x dx = x\arccos x-\sqrt{1-x^2}+C
Indefinite Integral
Trigonometry
68.
Integral of arctan
∫arctan⁡xdx=xarctan⁡x−12ln⁡(x2+1)+C\int \arctan x dx = x\arctan x - \dfrac{1}{2}\ln\left(x^2+1\right)+C
Indefinite Integral
Trigonometry
69.
∫ax+bdx=23a(ax+b)3/2+C\int \sqrt{ax+b}dx=\dfrac{2}{3a}\left(ax+b\right)^{3/2}+C
Indefinite Integral
Irrational Function
70.
∫xax+bdx=2(3ax−2b)15a2(ax+b)3/2+C\int x\sqrt{ax+b}dx=\dfrac{2(3ax-2b)}{15a^2}(ax+b)^{3/2}+C
Indefinite Integral
Irrational Function
71.
∫dx(x+c)ax+b=1b−acln⁡∣ax+b−b−acax+b+b−ac∣+C,b−ac>0\int \dfrac{dx}{(x+c)\sqrt{ax+b}}=\dfrac{1}{\sqrt{b-ac}} \ln\left|\dfrac{\sqrt{ax+b}-\sqrt{b-ac}}{\sqrt{ax+b}+\sqrt{b-ac}}\right|+C, b-ac>0
Indefinite Integral
Irrational Function
72.
∫dx(x+c)ax+b=1ac−barctan⁡ax+bac−b+C,b−ac<0\int \dfrac{dx}{(x+c)\sqrt{ax+b}}=\dfrac{1}{\sqrt{ac-b}}\arctan\sqrt{\dfrac{ax+b}{ac-b}}+C, b-ac<0
Indefinite Integral
Irrational Function
73.
∫x2a+bxdx=2(8a2−12abx+15b2x2)105b3(a+bx)3+C\int x^2\sqrt{a+bx}dx=\dfrac{2\left(8a^2-12abx+15b^2x^2\right)}{105b^3}\sqrt{(a+bx)^3}+C
Indefinite Integral
Irrational Function
74.
∫x2a+bx=2(8a2−4abx+3b2x2)15b3a+bx+C\int \dfrac{x^2}{\sqrt{a+bx}} = \dfrac{2\left(8a^2-4abx+3b^2x^2\right)}{15b^3}\sqrt{a+bx}+C
Indefinite Integral
Irrational Function
75.
∫exdx=ex+C\int e^x dx = e^x + C
E
Indefinite Integral
76.
∫axdx=axln⁡a+C \int a^x dx = \dfrac{a^x}{\ln a} + C
Exponents
Indefinite Integral
77.
∫eaxdx=eaxa+C \int e^{ax} dx = \dfrac{e^{ax}}{a}+C
Exponents
Indefinite Integral
78.
∫xeaxdx=eaxa2(ax−1)+C \int x e^{ax}dx = \dfrac{e^{ax}}{a^2}(ax-1)+C
E
Indefinite Integral
79.
∫ln⁡xdx=xln⁡x−x+C \int \ln x dx = x\ln x - x + C
Indefinite Integral
Logarithm
80.
∫dxxln⁡x=ln⁡∣ln⁡x∣+C \int \dfrac{dx}{x\ln x} = \ln|\ln x| + C
Indefinite Integral
Logarithm

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