Skip to main content

Integral with x2+a2\sqrt{x^2+a^2}

31​

∫1x2+a2 dx=sinh⁡xa+C=ln⁡(x+x2+a2)+C\int{\frac{1}{\sqrt{x^2+a^2}}}\ \mathrm{d}x=\sinh{\frac{x}{a}}+C= \ln{\left(x+\sqrt{x^2+a^2}\right)}+C
Derivation

32​

∫1(x2+a2)3 dx=xa2x2+a2+C\int{\frac{1}{\sqrt{(x^2+a^2)^3}}}\ \mathrm{d}x=\frac{x}{a^2\sqrt{x^2+a^2}}+C
Derivation

33​

∫xx2+a2 dx=x2+a2+C\int{\frac{x}{\sqrt{x^2+a^2}}}\ \mathrm{d}x=\sqrt{x^2+a^2}+C
Derivation

34​

∫x(x2+a2)3 dx=−1x2+a2+C\int{\frac{x}{\sqrt{(x^2+a^2)^3}}}\ \mathrm{d}x=-\frac{1}{\sqrt{x^2+a^2}}+C
Derivation

35​

∫x2x2+a2 dx=x2x2+a2−a22ln⁡(x+x2+a2)+C\int{\frac{x^2}{\sqrt{x^2+a^2}}}\ \mathrm{d}x= \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln{\left(x+\sqrt{x^2+a^2} \right)}+C
Derivation

36​

∫x2(x2+a2)3 dx=−xx2+a2+ln⁡(x+x2+a2)+C\int{\frac{x^2}{\sqrt{(x^2+a^2)^3}}}\ \mathrm{d}x= -\frac{x}{\sqrt{x^2+a^2}}+\ln{\left(x+\sqrt{x^2+a^2} \right)}+C
Derivation

37​

∫1xx2+a2 dx=1aln⁡x2+a2−a∣x∣+C\int{\frac{1}{x\sqrt{x^2+a^2}}}\ \mathrm{d}x=\frac{1}{a}\ln{\frac{\sqrt{x^2+a^2}-a}{\left|x\right|}}+C
Derivation

38​

∫1x2x2+a2 dx=−x2+a2a2x+C\int{\frac{1}{x^2\sqrt{x^2+a^2}}}\ \mathrm{d}x= -\frac{\sqrt{x^2+a^2}}{a^2x}+C
Derivation

39​

∫x2+a2 dx=x2x2+a2+a22ln⁡(x+x2+a2)+C\int{\sqrt{x^2+a^2}}\ \mathrm{d}x= \frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln{\left(x+\sqrt{x^2+a^2} \right)}+C
Derivation

40​

∫(x2+a2)3 dx=x8(2x2+5a2)x2+a2+38a4ln⁡(x+x2+a2)+C\int{\sqrt{(x^2+a^2)^3}}\ \mathrm{d}x=\frac{x}{8}(2x^2+5a^2)\sqrt{x^2+a^2}+\frac{3}{8}a^4\ln{\left(x+\sqrt{x^2+a^2} \right)}+C
Derivation

41​

∫xx2+a2 dx=13(x2+a2)3+C\int{x\sqrt{x^2+a^2}}\ \mathrm{d}x=\frac{1}{3}\sqrt{(x^2+a^2)^3}+C
Derivation

42​

∫x2x2+a2 dx=x8(2x2+a2)x2+a2−a48ln⁡(x+x2+a2)+C\int{x^2\sqrt{x^2+a^2}}\ \mathrm{d}x=\frac{x}{8}(2x^2+a^2)\sqrt{x^2+a^2}-\frac{a^4}{8}\ln{\left(x+\sqrt{x^2+a^2} \right)}+C
Derivation

43​

∫x2+a2x dx=x2+a2+aln⁡x2+a2−a∣x∣+C\int{\frac{\sqrt{x^2+a^2}}{x}}\ \mathrm{d}x=\sqrt{x^2+a^2}+a\ln{\frac{\sqrt{x^2+a^2}-a}{\left|x\right|}}+C
Derivation

44​

∫x2+a2x2 dx=−x2+a2x+ln⁡(x+x2+a2)+C\int{\frac{\sqrt{x^2+a^2}}{x^2}}\ \mathrm{d}x=-\frac{\sqrt{x^2+a^2}}{x}+\ln{\left(x+\sqrt{x^2+a^2} \right)}+C
Derivation