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Intuitive Calculus: Integration Exercises and Substitutions, Exercises of Calculus

Kent State University (KSU) - Ashtabula CampusCalculus

This document from ksu's math 11012 course provides integration exercises with various substitutions for evaluating indefinite integrals. Students are asked to find differentials, perform substitutions, and check their answers by differentiating.

Typology: Exercises

2019/2020

Uploaded on 06/18/2020

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alfred67 ๐Ÿ‡บ๐Ÿ‡ธ

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MATH 11012 Intuitive Calculus KSU
Integration Exercises
1. Find each of the following differentials.
(a) For y=f(x) = x100, find dy.
(b) For u=g(x) = ex, find du.
(c) For u=g(x) = 5ex+x, find du.
(d) For u=h(x) = ln(x2
โˆ’1), find du.
2. Find each indefinite integral using the substitution indicated. Check your answer by differentiating.
(a) Evaluate Z3x2
x3
โˆ’4dx using the substitution u=x3
โˆ’4.
(b) Evaluate Z3x2
(x3โˆ’4)5dx using the substitution u=x3
โˆ’4.
(c) Evaluate Zln(xโˆ’2)
xโˆ’2dx using the substitution u= ln(xโˆ’2).
(d) Evaluate Zx4ex5dx using the substitution u=x5.
(e) Evaluate Z1
1โˆ’5xdx using the substitution u= 1 โˆ’5x.
(f) Evaluate Zx4(x5+ 7)โˆ’3dx using the substitution u=x5+ 7.
3. Evaluate each indefinite integral. Try simplifying the integrand algebraically instead of or in addition
to using a substitution. Check your answer by differentiating.
(a) Ze3x๎˜€2โˆ’e3x๎˜5dx
(b) Zโˆšln x
xdx
(c) Z1
xโˆšln xdx
(d) Zx2
โˆ’36
x+ 6 dx
(e) Zeโˆ’1/x
x2dx
(f) Zx3(x+ 3)2dx
(g) Zex+ 1
ex+xdx
(h) Ze2x+x
e2x+x2dx
(i) Z1
xln xdx
(j) Z(x+ 2)3
x2dx
(k) Zexโˆšexโˆ’17 dx
(l) Zx
x2
โˆ’5dx
(m) Zx2
โˆ’5
xdx
(n) Zx ex2
โˆ’1dx
(o) Zx ex2
โˆ’2xโˆ’ex2
โˆ’2xdx
1

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MATH 11012 Intuitive Calculus KSU

Integration Exercises

  1. Find each of the following differentials.

(a) For y = f (x) = x 100 , find dy.

(b) For u = g(x) = e x , find du.

(c) For u = g(x) = 5e x

  • x, find du.

(d) For u = h(x) = ln(x 2 โˆ’ 1), find du.

  1. Find each indefinite integral using the substitution indicated. Check your answer by differentiating.

(a) Evaluate

3 x^2

x^3 โˆ’ 4

dx using the substitution u = x 3 โˆ’ 4.

(b) Evaluate

3 x^2

(x^3 โˆ’ 4)^5

dx using the substitution u = x 3 โˆ’ 4.

(c) Evaluate

ln(x โˆ’ 2)

x โˆ’ 2

dx using the substitution u = ln(x โˆ’ 2).

(d) Evaluate

x 4 e x^5 dx using the substitution u = x 5 .

(e) Evaluate

1 โˆ’ 5 x

dx using the substitution u = 1 โˆ’ 5 x.

(f) Evaluate

x 4 (x 5

โˆ’ 3 dx using the substitution u = x 5

  1. Evaluate each indefinite integral. Try simplifying the integrand algebraically instead of or in addition to using a substitution. Check your answer by differentiating.

(a)

e 3 x ( 2 โˆ’ e 3 x)^5 dx

(b)

ln x

x

dx

(c)

x

ln x

dx

(d)

x^2 โˆ’ 36

x + 6

dx

(e)

eโˆ’^1 /x

x^2

dx

(f)

x 3 (x + 3) 2 dx

(g)

e x

  • 1

ex^ + x

dx

(h)

e 2 x

  • x

e^2 x^ + x^2

dx

(i)

x ln x

dx

(j)

(x + 2)^3

x^2

dx

(k)

e x

ex^ โˆ’ 17 dx

(l)

x

x^2 โˆ’ 5

dx

(m)

x^2 โˆ’ 5

x

dx

(n)

x e x^2 โˆ’ 1 dx

(o)

x e x^2 โˆ’ 2 x โˆ’ e x^2 โˆ’ 2 x dx