Techniques: substitution

Practice sheet

Fresh questions from the whole chapter, to do on paper.

New questions

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  1. Find FF with F′(x)=2xcos⁡(x2)+4F'(x) = 2x\cos(x^2) + 4 and F(0)=0F(0) = 0.
  2. Find FF with F′(x)=x ex2F'(x) = x\,e^{x^2} and F(0)=3F(0) = 3.
  3. Find FF with F′(x)=(5x+2)2F'(x) = (5x + 2)^{ 2 } and F(0)=0F(0) = 0.
  4. Evaluate ∫012x(x2+1)4 dx\displaystyle\int_0^1 2x(x^2 + 1)^{ 4 }\,dx.
  5. Evaluate ∫0π/2sin⁡6xcos⁡x dx\displaystyle\int_0^{\pi/2} \sin^{ 6 } x \cos x\,dx.
  6. Evaluate ∫1e(ln⁡x)1x dx\displaystyle\int_1^{e} \dfrac{(\ln x)^{ 1 } }{x}\,dx.
  7. Evaluate ∫012xx2+2 dx\displaystyle\int_0^1 \dfrac{2x}{x^2 + 2}\,dx.
  8. Find FF with F′(x)=tan⁡xF'(x) = \tan x on (−π2,π2)\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right) and F(0)=3F(0) = 3.
  9. Find FF with F′(x)=e2xF'(x) = e^{ 2x} and F(0)=0F(0) = 0.
  10. Find FF with F′(x)=1x2+9F'(x) = \dfrac{1}{x^2 + 9} and F(0)=0F(0) = 0.
  11. Find FF with F′(x)=1x2+6x+10F'(x) = \dfrac{1}{x^2 + 6x + 10} and F(−3)=0F(-3) = 0.
  12. Evaluate ∫12x2+5x+1x dx\displaystyle\int_1^2 \dfrac{x^2 + 5x + 1}{x}\,dx.

Answers

  1. 4x+sin⁡(x2)4 x + \sin{\left(x^{2} \right)}
  2. ex22+52\frac{e^{x^{2}}}{2} + \frac{5}{2}
  3. (5x+2)315−815\frac{\left(5 x + 2\right)^{3}}{15} - \frac{8}{15}
  4. 315\frac{31}{5}
  5. 17\frac{1}{7}
  6. 12\frac{1}{2}
  7. ln⁡(32)\ln{\left(\frac{3}{2} \right)}
  8. 3−ln⁡(cos⁡(x))3 - \ln{\left(\cos{\left(x \right)} \right)}
  9. e2x2−12\frac{e^{2 x}}{2} - \frac{1}{2}
  10. tan⁡−1(x3)3\frac{\tan^{-1}{\left(\frac{x}{3} \right)}}{3}
  11. tan⁡−1(x+3)\tan^{-1}{\left(x + 3 \right)}
  12. ln⁡(2)+132\ln{\left(2 \right)} + \frac{13}{2}