Problems from Integrals of functions defined by parts

Development:

$$\displaystyle \int_0^2 (x+e^x) \ dx + \int_2^3 3 \ dx + \int_3^4 (2+e^{2x}) \ dx =$$

$$= \Big[\dfrac{x^2}{2}+e^x\Big]^2_0 + [3x]^3_2 + [2x]^4_3 + [\frac{1}{2}e^{2x}]^4_3=$$

$$=(2+e^2-1)+(9-6)+(8-6)+(\frac{1}{2}e^8-\frac{1}{2}e^6)=$$

$$=6+e^2+\frac{1}{2}(e^8-e^6)$$

Solution:

The result is $$6+e^2+\frac{1}{2}(e^8-e^6)$$

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Development:

The function $$f(x)=|x|$$ can be expressed as a piecewise function as follows: $$$\displaystyle |x|=\left\{\begin{array}{rcl} -x & \mbox{ if } & x<0 \\ x & \mbox{if} & x\geq0 \end{array}\right.$$$

Let's solve the integral piecewise

$$\displaystyle \int_{-1}^0 (-x) \ dx + \int_0^1 x \ dx =$$

$$= \Big[\dfrac{-x^2}{2}\Big]^0_{-1} + \Big[\dfrac{x^2}{2}\Big]^1_0=$$

$$=0-\Big(-\dfrac{1}{2}\Big)+\dfrac{1}{2}-0=1$$

Solution:

The result is $$1$$

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