What is the integral of e^{(x+5)^7} ?

The integral of e^{(x+5)^7} is -((x+5)\Gamma(1/7 ,-(x+5)^7))/(7\sqrt[7]{-(x+5)^7)}+C

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integral of e^{(x+5)^7}

\int e^{(x + 5)^{7}} d x

- \frac{( x + 5 ) Γ ( \frac{1}{7} , - ( x + 5 ) ^{7} )}{7 7 - ( x + 5 ) ^{7}} + C

Solution steps

\int e^{(x + 5)^{7}} d x

Apply u-substitution

: \int e^{u^{7}} d u

= \int e^{u^{7}} d u

Use the nonelementary integral:

\int e^{x^{a}} d x = - \frac{x Γ ( \frac{1}{a} , - x ^{a} )}{a a - x ^{a}}

= - \frac{u Γ ( \frac{1}{7} , - u ^{7} )}{7 7 - u ^{7}}

Substitute back

u = x + 5

= - \frac{( x + 5 ) Γ ( \frac{1}{7} , - ( x + 5 ) ^{7} )}{7 7 - ( x + 5 ) ^{7}}

Add a constant to the solution

= - \frac{( x + 5 ) Γ ( \frac{1}{7} , - ( x + 5 ) ^{7} )}{7 7 - ( x + 5 ) ^{7}} + C

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What is the integral of e^{(x+5)^7} ?
The integral of e^{(x+5)^7} is -((x+5)\Gamma(1/7 ,-(x+5)^7))/(7\sqrt[7]{-(x+5)^7)}+C