Integral of \( \displaystyle e^{x} \cos{\left(x \right)} \)

Problem 4.664 · easy

1. \[ \int e^{x} \cos{\left(x \right)}\, dx \]
   integralStart with the integral of the integrand.✓ Proved
2. \[ = \int \frac{\left(e^{i x} + e^{- i x}\right) e^{x}}{2}\, dx \]
   trig-identityUse the exponential form of the cosine function.≈ Checked numerically
3. \[ = \int \frac{e^{x} e^{- i x}}{2}\, dx + \int \frac{e^{x} e^{i x}}{2}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = \int \frac{e^{x \left(1 - i\right)}}{2}\, dx + \int \frac{e^{x \left(1 + i\right)}}{2}\, dx \]
   algebraCombine the exponents.✓ Proved
5. \[ = \frac{\left(2 + 2 i\right) e^{x \left(1 - i\right)}}{8} + \frac{\left(2 - 2 i\right) e^{x \left(1 + i\right)}}{8} \]
   antiderivativeIntegrate each exponential term.✓ Proved
6. \[ = \frac{\left(2 - 2 i\right) e^{x} e^{i x}}{8} + \frac{\left(2 + 2 i\right) e^{x} e^{- i x}}{8} \]
   algebraSeparate the exponential terms.✓ Proved
7. \[ = \left(\frac{\left(2 - 2 i\right) e^{i x}}{8} + \frac{\left(2 + 2 i\right) e^{- i x}}{8}\right) e^{x} \]
   algebraFactor out exp(x).✓ Proved
8. \[ = \frac{\left(1 + i\right) \left(2 - 2 i\right) \left(\left(1 - i\right) e^{i x} + \left(1 + i\right) e^{- i x}\right) e^{x}}{16} \]
   algebraFind a common denominator.✓ Proved
9. \[ = \frac{\left(\left(1 - i\right) e^{i x} + \left(1 + i\right) e^{- i x}\right) e^{x}}{4} \]
   algebra algebraExpand the denominator. Simplify the denominator.✓ Proved
10. \[ = \frac{\left(e^{i x} - i e^{i x} + e^{- i x} + i e^{- i x}\right) e^{x}}{4} \]
    algebraDistribute the terms in the numerator.✓ Proved
11. \[ = \frac{\left(- i \left(e^{i x} - e^{- i x}\right) + e^{i x} + e^{- i x}\right) e^{x}}{4} \]
    algebraGroup the terms to form cosine and sine components.✓ Proved
12. \[ = \frac{\left(2 \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{x}}{4} \]
    trig-identity algebraSubstitute the exponential forms of sine and cosine. Simplify the numerator by multiplying I*I.≈ Checked numerically
13. \[ = \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}}{2} \]
    simplifyFinal simplification.✓ Proved