5.5 Math Notes.pdf - c Amy Austin Section 5.5 Integration by Substitution The Substitution Rule If u = g(x is a differentiable function then Z \u2032

5.5 Math Notes.pdf - c Amy Austin Section 5.5 Integration...

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Unformatted text preview: c Amy Austin, January 14, 2019 Section 5.5: Integration by Substitution The Substitution Rule: If u = g(x) is a differentiable function, then Z ′ f (g(x))g (x) dx = Z f (u) du Note: Typically, u is chosen so that du is a factor of the integrand. 1. Z 2t2 (t3 − 1)3 dt 2. Z √ sec2 x dx tan x + 9 1 c Amy Austin, January 14, 2019 3. Z 0 13 1 dx (1 + 2x)2 p 3 4. Z x2 dx (1 − x)4 5. Z e1/x dx x2 2 c Amy Austin, January 14, 2019 6. Z π/4 esin(2t) cos(2t) dt 0 7. Z tan(x) dx 8. Z arctan x dx 1 + x2 3 c Amy Austin, January 14, 2019 9. Z 10. Z x+1 dx x2 + 1 e4 e3 1 dx x ln x Note: If the choice of u is linear (degree 1), then Zdu is a constant multiple of dx and hence can be divided out of the integral. To illustrate, let’s find ekx dx. 4 c Amy Austin, January 14, 2019 3 − 4x − 1 11. Z  12. Z (sin(3α) − sin(3x)) dx 13. Z √ 14. Z x dx 1 + x4 −10x cos(5x) + e  dx sin x dx 1 − cos2 x 5 ...
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