SOLVED: 5-18 Evaluate the definite integral two ways: first by a u-substitution in the definite integral and then by a u-substitution in the corresponding indefinite integral. âˆ«(x + 1) dx âˆ«(4x

U-substitution with definite integral
U-substitution with definite integral

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5-18 Evaluate the definite integral two ways: first by a u-substitution in the definite integral and then by a u-substitution in the corresponding indefinite integral.
âˆ«(x + 1) dx
âˆ«(4x – 2)^3 dx
âˆ«(2x – 1)^3 dx
âˆ«(4 – 3x)^8 dx
âˆ«(x^2 + x) dx
âˆ«(x^2 – x) dx
âˆ«sin(x/2) dx
âˆ«cos(3x) dx
âˆ«(6x^2 + 25) dx
âˆ«sec^2(x – 4) dx
âˆ«ln(3x) dx
âˆ«ln(3x^2 + 4e^3) dx
âˆ«(6/x) ln(2/âˆš3) dx
âˆ«e^(-2x) dx

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01:07

Exercise 65: Evaluate the following integrals.1. âˆ«xâˆš(9 – x^2) dxAns: xâˆš(9 – x^2) + C2. âˆ«3 dxAns: 3x + C3. âˆ«dy/âˆš(1 + y^2)Ans: ln|y + âˆš(1 + y^2)| + C4. âˆ«(1/âˆš(25 + 9y^2)) dyAns: (1/3)ln|3y + âˆš(25 + 9y^2)| + C5. âˆ«x/âˆš(4 – x^2) dxAns: -âˆš(4 – x^2) + C6. âˆ«y/âˆš(1 – 3y^2 + 9y^4) dyAns: (1/6)sin^(-1)(3y^2) + C7. âˆ«e^x/âˆš(1 – e^(2x)) dxAns: sin^(-1)(e^x) + C8. âˆ«(2 + y)/âˆš(2 + y)âˆš(y) dyAns: (2/3)tan^(-1)(âˆšy) + C9. âˆ«âˆš(2x – x^2) dxAns: xâˆš(2 – x) + C10. âˆ«(25 + 16x)/(7x) dxAns: 25ln|7x| + 16x + C11. âˆ«(9 + 77x)/(x^2) dxAns: -9/x + 77ln|x| + C12. âˆ«(x^2 + 9)/(x^2) dxAns: x + 9ln|x| + C13. âˆ«(4 + 5x)/(1 + x^2) dxAns: 4tan^(-1)(x) + 5ln|1 + x^2| + C

01:20

Evaluating a Definite Integral In Exercises 9-34, evaluate the definite integral. Use a graphing utility to verify your result:âˆ«(2x + 1) dx from 1 to 3âˆ«(t^2 – 1) dt from -1 to 1âˆ«(6 – âˆš(x)) dx from -1 to 1âˆ«(y^2) du from 0 to 1âˆ«(f'(w – 2)) dt from 0 to 1âˆ«(5âˆš(6x – 4)) dt from 1 to 2âˆ«(5âˆ›(12/3)) dt from 0 to 1âˆ«((6r^2 – 3x)) dx from 0 to 1âˆ«(âˆš(r^2 – x)) dx from 0 to 1âˆ«(6 – 4u) du from 0 to 1âˆ«(xâˆ›(3)) dx from 0 to 1âˆ«(2x + 5) dx from 0 to 1âˆ«(r^2 – 4r + 3) dx from 0 to 1âˆ«((sin x + 7)) dx from 0 to 1âˆ«((2 + cos x)) dx from 0 to 1âˆ«(sec^2(Î¸)) dÎ¸ from 0 to Ï€/4âˆ«(sin^2(Î¸)) dÎ¸ from 0 to Ï€/2âˆ«(3/6sec^2(x)) dx from -Ï€/6 to Ï€/6âˆ«(7/2csc^2(x)) dx from 0 to Ï€/4

04:11

Consider the integral âˆ«x^4 (x^5-10)^6 dx.To find the value of this integral, make thesubstitution u=x^5 – 10.a) Write (x^5-10)^6 in terms of u: b) This makes x^4 dx = c) Re-write the integral in terms of u and du, andfind the antiderivative.The antiderivative in termsof u is: d) Now back-substitute and write your answer to part (c) in termsof x:

02:20

9) âˆ«(x^2 – dx) = -5A) – 4 sin(x) – âˆš10 + CB) -24 sin(x) – âˆš10 + CC) 5 + sin(3x) + âˆš10 + CD) 3 + âˆš(sin(x) + C)

10) Evaluate the integral: âˆ«sin(t)(4+cos(t))^3A) 204 + cos(t)^2B) âˆ«4(4+cos(t))^4 + CC) âˆ«(4+cos(t))^2 + CD) âˆ«(4+cos(t))^2 + C

01:00

Evaluate the following integrals

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