5.3 The fundamental theorem of calculus; Questions 148-159

[Eigenvalue]

Us the fundamental theorem of calculus, part 1, to find each derivative
148.[tex]\frac{d}{dx} \int\limits_{1}^{x} e^{t² }[/tex]dt

149.[tex]\frac{d}{dx} \int\limits_{1}^{x} e^{cos(t)}[/tex]dt

150. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{3}^{x} \sqrt{9-y²}[/tex]dy

151. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{3}^{x}[/tex] [tex]\frac{ds}{ \sqrt{16-s²} }[/tex]

152. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{x}^{2x}[/tex](t)dt

156. [tex]\frac{d}{dx} \int\limits_{0}^{ \sqrt{x} }[/tex] [tex]\frac{t²}{1+t⁴}[/tex]dt

154. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{0}^{sin(x)}[/tex] [tex]\sqrt{1-t²}[/tex]dt

155. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{1}^{cos(x)}[/tex] [tex]\sqrt{1-t²}[/tex]dt

153. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{1}^{ \sqrt{x} }[/tex](t)dt

157. [tex]\frac{d}{dx} \int\limits_{1}^{x²} \frac{ \sqrt{t} }{1+t}[/tex]dt

158. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{0}^{ln(x)} e^{t }[/tex]

159. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{1}^{ e^{x } }[/tex] ln(u²)du

[Eigenvalue]

148.[tex]\frac{d}{dx} \int\limits_{1}^{x} e^{t² }[/tex]dt
Substitute x for t:
[tex]e^{x² }[/tex]

[Eigenvalue]

149.[tex]\frac{d}{dx} \int\limits_{1}^{x} e^{cos(t)}[/tex]dt
Substitute x for t:
[tex]e^{cos(x)}[/tex]

[Eigenvalue]

150. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{3}^{x} \sqrt{9-y²}[/tex]dy
Substitute x for y:
[tex]\sqrt{9-x²}[/tex]

[Eigenvalue]

151. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{3}^{x}[/tex] [tex]\frac{ds}{ \sqrt{16-s²} }[/tex]
Substitute x for s:
[tex]\frac{1}{ \sqrt{16-x²} }[/tex]
Rationalize the denominator: [tex]\frac{ \sqrt{16-x²} }{16-x²}[/tex]

[Eigenvalue]

152. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{x}^{2x}[/tex](t)dt

[tex]\int\limits_{2x}^{1}[/tex](t)dt +[tex]\int\limits_{1}^{x}[/tex](t)dt
=[tex]\int\limits_{1}^{2x}[/tex](t)dt -[tex]\int\limits_{x}^{1}[/tex](t)dt
=2(2x)-x
=3x

[Eigenvalue]

156. [tex]\frac{d}{dx} \int\limits_{0}^{ \sqrt{x} }[/tex] [tex]\frac{t²}{1+t⁴}[/tex]dt

=([tex]\sqrt{x}[/tex])'[tex]\frac{( \sqrt{x})² }{1+( \sqrt{x})⁴ }[/tex]
=[tex]\frac{1}{2} x^{-1/2} \frac{x}{1+x²}[/tex]
=[tex]\frac{ \sqrt{x} }{2+2x²}[/tex]

[Eigenvalue]

154. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{0}^{sin(x)}[/tex] [tex]\sqrt{1-t²}[/tex]dt

(sin(x))'[tex]\sqrt{1-(sin²(x)}[/tex]
=cos(x)[tex]\sqrt{cos²(x)}[/tex]
=cos(x)|cos(x)|

[Eigenvalue]

155. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{1}^{cos(x)}[/tex] [tex]\sqrt{1-t²}[/tex]dt

(cos(x))'[tex]\sqrt{1-cos²(x)}[/tex]
=-sin(x)[tex]\sqrt{sin²(x)}[/tex]
=-sin(x)|sin(x)|

[Eigenvalue]

153. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{1}^{ \sqrt{x} }[/tex](t)dt

([tex]\sqrt{x}[/tex])'([tex]\sqrt{x}[/tex])
=[tex]\frac{1}{2} x^{-1/2 }[/tex]*[tex]x^{1/2 }[/tex]

=[tex]\frac{1}{2}[/tex]

[Eigenvalue]

157. [tex]\frac{d}{dx} \int\limits_{1}^{x²} \frac{ \sqrt{t} }{1+t}[/tex]dt

(x²)'([tex]\frac{ \sqrt{x²} }{1+x²}[/tex])

=2x([tex]\frac{|x|}{1+x²}[/tex])

=[tex]\frac{2x|x|}{1+x²}[/tex]

[Eigenvalue]

158. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{0}^{ln(x)} e^{t }[/tex]

(ln(x))'([tex]e^{ln(x) }[/tex])

=[tex]\frac{1}{x}[/tex]*x
=1

[Eigenvalue]

159. [tex]\frac{d}{dx}[/tex] [tex]\int\limits_{1}^{ e^{x } }[/tex] ln(u²)du

([tex]e^{x}[/tex])(ln([tex]e^{x }[/tex])²)

=[tex]e^{x }[/tex](2ln([tex]e^{x}[/tex]))
=2x[tex]e^{x}[/tex]