`int_0^1 (x^10 + 10^x) dx` Evaluate the integral

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Chapter 5, 5.4 - Problem 35 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the integral, hence, you need to use the fundamental theorem of calculus, such that:

`int_a^b f(x)dx = F(b) - F(a)`

`int_0^1(x^10 + 10^x)dx = int_0^1(x^10)dx + int_0^1 10^x dx`

Evaluating each definite integral, yields:

`int_0^1(x^10)dx = (x^10)/10|_0^1 = (1/10)(1^10 - 0^10) = 1/10`

`int_0^1 10^x dx = (10^x)/(ln 10)|_0^1 = (10^1)/(ln 10) - (10^0)/(ln 10)`

`int_0^1 10^x dx =1/(ln 10)(10 - 1) = 9/(ln 10)`

Hence, evaluating the definite integral, using the fundamental theorem of calculus yields `int_0^1(x^10 + 10^x)dx = 1/10 + 9/(ln 10).`

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