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u-Substitution Recall the substitution rule from MATH 141 (see page 241 in the textbook). Theorem If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ˆ f(g(x))g′(x)dx = ˆ f(u)du. This method of integration is helpful in reversing the chain rule (Can you see why?) Let’s look at some ...
Integration Worksheet - Substitution Method Solutions (a)Let u= 4x 5 (b)Then du= 4 dxor 1 4 du= dx (c)Now substitute Z p 4x 5 dx = Z u 1 4 du = Z 1 4 u1=2 du 1 4 u3=2 2 3 +C = 1
AP Calculus—Integration Practice I. Integration by substitition. Basic Idea: If u= f(x), then du= f0(x)dx: Example. We have Z xdx x4 +1 u= x2 = dx= 2xdx 1 2 Z du u2 +1 = 1 2 tan 1 u+C = 1 2 tan 1 x2 +C Practice Problems: 1. Z x3 p 4+x4 dx 2. Z dx xlnx 3. Z (x+5)dx p x+4 4. In each integral below, find the integer nthat allows for an ...
Answer the following questions by using the integration technique known as u-substitution. \(\displaystyle \int (2x+1)^2dx\) \(\displaystyle \int x\sqrt{2x+3}dx\)
u-substitution works for integrating compositions of functions; pick u to be the ’inside’ function (for inde nite integrals, drop the limits of integration). 1.
Math 229 Integration Worksheet – Substitution Method Integrate 1. Z (5x+4)5 dx 2. Z 3t2(t3 +4)5 dt 3. Z p 4x5dx 4. Z t2(t3 +4)1/2 dt 5. Z cos(2x+1)dx 6. Z sin10 xcosxdx 7. Z sinx (cosx)5 dx 8. Z (p x1)2 p x dx 9. Z p x3 +x2(3x2 +2x)dx 10. Z 1 1 x+1 (x2 +2x+2)3 dx 11. Z ⇡ 0 cosx p sinxdx 12. Z (x+1)sin(x2 +2x+3)dx 13. Z ⇣ 1+ 1 t ⌘ 3 1 t2 ...
Math 141: u-Substitution Practice Warm-Up/Review Problems 1. Evaluate the following integrals (a) Z x2(p x+ 5) + e2dx (b) Z 2 1 3x3 + 1 4x dx 2. Find the area bounded between the line f(x) = x + 3 and the parabola g(x) = x2 + x+ 2. (a) Find where f(x) and g(x) intersect by setting them equal and solving for x.
