such as find an antiderivative for 3sinx and and e 3x and i

such as find an antiderivative for 3sin(x), and , and e 3x and in the form off 8y-21 ) dy. 30. Evaluate the following definite integrals (any and all from homework and class examples). NOTE: There will be an application like the ones presented in class.

Solution

a) _{[1 to 2]} x dx = _{[1 to 2]} x²/2          since xⁿ dx = xⁿ⁺¹/(n +1)

= (2)²/2 - (1)²/2 = 1/2

b) _{[0 to /2]} ² d = _{[0 to /2]} ³/3

= (/2)³/3 - (0)³/3

= ³/24

c) _{[0 to 2]} 3x² + x - 5 dx

= _{[0 to 2]} 3x³/3 + x²/2 - 5x

= _{[0 to 2]} x³ + x²/2 - 5x

= (2³ + 2²/2 - 5(2)) - (0³ + 0²/2 - 5(0))

= 0

d) _{[0 to b^1/3]} x² dx

= _{[0 to b^1/3]} x³/3

= ((b^1/3)³/3) - (0³/3)

= b/3

e) _{[0 to 1/2]} 4/(1 -x²) dx

= _{[0 to 1/2]} 4sin^-1x               since 1/(1 -x²) dx = sin^-1x

= 4[sin^-1(1/2) - sin^-1(0) ]

= 4[/6 - 0]

= 2/3

f) _{[2 to -4]} x^-1 dx

= _{[2 to -4]} x^-1+1/(-1+1)

= _{[2 to -4]} x/()

= (1/)((-4) -(2))