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Integral with Inverse Trig Functions

113​

∫arcsin⁡xa dx=xarcsin⁡xa+a2−x2+C\int{\arcsin{\frac{x}{a}}}\ \mathrm{d}x=x\arcsin{\frac{x}{a}}+\sqrt{a^2-x^2}+C
Derivation

114​

∫xarcsin⁡xa dx=(x22−a24)arcsin⁡xa+x4a2−x2+C\int{x\arcsin{\frac{x}{a}}}\ \mathrm{d}x=\left(\frac{x^2}{2}-\frac{a^2}{4} \right) \arcsin{\frac{x}{a}}+\frac{x}{4}\sqrt{a^2-x^2}+C
Derivation

115​

∫x2arcsin⁡xa dx=x33arcsin⁡xa+19(x2+2a2)a2−x2+C\int{x^2\arcsin{\frac{x}{a}}}\ \mathrm{d}x=\frac{x^3}{3}\arcsin{\frac{x}{a}}+\frac{1}{9}(x^2+2a^2)\sqrt{a^2-x^2}+C
Derivation

116​

∫arccos⁡xa dx=xarccos⁡xa−a2−x2+C\int{\arccos{\frac{x}{a}}}\ \mathrm{d}x=x\arccos{\frac{x}{a}}-\sqrt{a^2-x^2}+C
Derivation

117​

∫xarccos⁡xa dx=(x22−a24)arccos⁡xa+x4a2−x2+C\int{x\arccos{\frac{x}{a}}}\ \mathrm{d}x=\left(\frac{x^2}{2}-\frac{a^2}{4} \right) \arccos{\frac{x}{a}}+\frac{x}{4}\sqrt{a^2-x^2}+C
Derivation

118​

∫x2arccos⁡xa dx=x33arccos⁡xa−19(x2+2a2)a2−x2+C\int{x^2\arccos{\frac{x}{a}}}\ \mathrm{d}x=\frac{x^3}{3}\arccos{\frac{x}{a}}-\frac{1}{9}(x^2+2a^2)\sqrt{a^2-x^2}+C
Derivation

119​

∫arctan⁡xa dx=xarctan⁡xa−a2ln⁡(a2+x2)+C\int{\arctan{\frac{x}{a}}}\ \mathrm{d}x=x\arctan{\frac{x}{a}}-\frac{a}{2}\ln{(a^2+x^2)} +C
Derivation

120​

∫xarctan⁡xa dx=12(a2+x2)arctan⁡xa−a2x+C\int{x\arctan{\frac{x}{a}}}\ \mathrm{d}x=\frac{1}{2} (a^2+x^2) \arctan{\frac{x}{a}}-\frac{a}{2}x+C
Derivation

121​

∫x2arctan⁡xa dx=x33arctan⁡xa−a6x2+a36ln⁡(a2+x2)+C\int{x^2\arctan{\frac{x}{a}}}\ \mathrm{d}x=\frac{x^3}{3}\arctan{\frac{x}{a}}-\frac{a}{6}x^2+\frac{a^3}{6}\ln{(a^2+x^2)}+C
Derivation