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Integral with a2−x2\sqrt{a^2-x^2}

59​

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

60​

∫1(a2−x2)3 dx=xa2a2−x2+C\int{\frac{1}{\sqrt{(a^2-x^2)^3}}}\ \mathrm{d}x=\frac{x}{a^2 \sqrt{a^2-x^2}}+C
Derivation

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∫xa2−x2 dx=−a2−x2+C\int{\frac{x}{\sqrt{a^2-x^2}}}\ \mathrm{d}x=-\sqrt{a^2-x^2}+C
Derivation

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

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∫x2a2−x2 dx=−x2a2−x2+a22arcsin⁡xa+C\int{\frac{x^2}{\sqrt{a^2-x^2}}}\ \mathrm{d}x=-\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin{\frac{x}{a}}+C
Derivation

64​

∫x2(a2−x2)3 dx=xa2−x2−arcsin⁡xa+C\int{\frac{x^2}{\sqrt{(a^2-x^2)^3}}}\ \mathrm{d}x=\frac{x}{\sqrt{a^2-x^2}}-\arcsin{\frac{x}{a}}+C
Derivation

65​

∫1xa2−x2 dx=1aln⁡a−a2−x2∣x∣+C\int{\frac{1}{x\sqrt{a^2-x^2}}}\ \mathrm{d}x=\frac{1}{a}\ln{\frac{a-\sqrt{a^2-x^2}}{\left|x\right|}}+C
Derivation

66​

∫1x2a2−x2 dx=−a2−x2a2x+C\int{\frac{1}{x^2\sqrt{a^2-x^2}}}\ \mathrm{d}x=-\frac{\sqrt{a^2-x^2}}{a^2x}+C
Derivation

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∫a2−x2 dx=x2a2−x2+a22arcsin⁡xa+C\int{\sqrt{a^2-x^2}}\ \mathrm{d}x=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin{\frac{x}{a}}+C
Derivation

68​

∫(a2−x2)3 dx=x8(5a2−2x2)a2−x2+38a4arcsin⁡xa+C\int{\sqrt{(a^2-x^2)^3}}\ \mathrm{d}x=\frac{x}{8}(5a^2-2x^2)\sqrt{a^2-x^2}+\frac{3}{8}a^4\arcsin{\frac{x}{a}}+C
Derivation

69​

∫xa2−x2 dx=−13(a2−x2)3+C\int{x\sqrt{a^2-x^2}}\ \mathrm{d}x =-\frac{1}{3}\sqrt{(a^2-x^2)^3}+C
Derivation

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∫x2a2−x2 dx=x8(2x2−a2)a2−x2+a48arcsin⁡xa+C\int{x^2\sqrt{a^2-x^2}}\ \mathrm{d}x =\frac{x}{8}(2x^2-a^2)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin{\frac{x}{a}}+C
Derivation

71​

∫a2−x2x dx=a2−x2+aln⁡a−a2−x2∣x∣+C\int{\frac{\sqrt{a^2-x^2}}{x}}\ \mathrm{d}x=\sqrt{a^2-x^2}+a\ln{\frac{a-\sqrt{a^2-x^2}}{\left|x\right|}}+C
Derivation

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∫a2−x2x2 dx=−a2−x2x+arcsin⁡xa+C\int{\frac{\sqrt{a^2-x^2}}{x^2}}\ \mathrm{d}x=-\frac{\sqrt{a^2-x^2}}{x}+\arcsin{\frac{x}{a}}+C
Derivation