Skip to main content

Integral with Exponential Functions

122​

∫ax dx=1ln⁡aax+C\int{a^{x}}\ \mathrm{d}x=\frac{1}{\ln{a}}a^{x}+C
Derivation

123​

∫eax dx=1aeax+C\int{e^{ax}}\ \mathrm{d}x=\frac{1}{a}e^{ax}+C
Derivation

124​

∫xeax dx=1a2(ax−1)eax+C\int{xe^{ax}}\ \mathrm{d}x=\frac{1}{a^2}(ax-1)e^{ax}+C
Derivation

125​

∫xneax dx=1axneax−na∫xn−1eax dx\int{x^{n}e^{ax}}\ \mathrm{d}x=\frac{1}{a}x^{n}e^{ax}-\frac{n}{a}\int{x^{n-1}e^{ax}}\ \mathrm{d}x
Derivation

126​

∫xax dx=xln⁡aax−1(ln⁡a)2ax+C\int{xa^{x}}\ \mathrm{d}x= \frac{x}{\ln{a}}a^x-\frac{1}{(\ln{a})^2}a^{x}+C
Derivation

127​

∫xnax dx=1ln⁡axnax−nln⁡a∫xn−1ax dx\int{x^{n}a^{x}}\ \mathrm{d}x=\frac{1}{\ln{a}}x^{n}a^{x}-\frac{n}{\ln{a}}\int{x^{n-1}a^{x}}\ \mathrm{d}x
Derivation

128​

∫eaxsin⁡bx dx=1a2+b2eax(asin⁡bx−bcos⁡bx)+C\int{e^{ax}\sin{bx}}\ \mathrm{d}x=\frac{1}{a^2+b^2}e^{ax}(a\sin{bx}-b\cos{bx})+C
Derivation

129​

∫eaxcos⁡bx dx=1a2+b2eax(acos⁡bx+bsin⁡bx)+C\int{e^{ax}\cos{bx}}\ \mathrm{d}x=\frac{1}{a^2+b^2}e^{ax}(a\cos{bx}+b\sin{bx})+C
Derivation

130​

∫eaxsin⁡nbx dx=1a2+b2n2eaxsin⁡n−1bx(asin⁡bx−nbcos⁡bx)+n(n−1)b2a2+b2n2∫eaxsin⁡n−2bx dx\int{e^{ax}\sin^n{bx}}\ \mathrm{d}x=\frac{1}{a^2+b^2n^2}e^{ax}\sin^{n-1}{bx}(a\sin{bx}-nb\cos{bx})+\frac{n(n-1)b^2}{a^2+b^2n^2}\int{e^{ax}\sin^{n-2}{bx}}\ \mathrm{d}x
Derivation

131​

∫eaxcos⁡nbx dx=1a2+b2n2eaxcos⁡n−1bx(acos⁡bx+nbsin⁡bx)+n(n−1)b2a2+b2n2∫eaxcos⁡n−2bx dx\int{e^{ax}\cos^n{bx}}\ \mathrm{d}x=\frac{1}{a^2+b^2n^2}e^{ax}\cos^{n-1}{bx}(a\cos{bx}+nb\sin{bx})+\frac{n(n-1)b^2}{a^2+b^2n^2}\int{e^{ax}\cos^{n-2}{bx}}\ \mathrm{d}x
Derivation