# Part 2: Advanced Integration Techniques

$$\newcommand{\sech}{\operatorname{sech}}$$ $$\newcommand{\csch}{\operatorname{csch}}$$ $$\newcommand{\inverse}[1]{#1^{\leftarrow}}$$

## 2.1 Integration by Substitution

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 5.5, 5.6

### 2.1.1 Substitution and the Chain Rule

• Reversing the Chain Rule $$\frac{d}{dx}[f(g(x))]=f’(g(x))g’(x)$$ yields the Substitution Rule $$\int f’(g(x))g’(x)\,dx=f(g(x))+C$$.
• This is often abbreviated as $$\int f’(u)\,du=f(u)+C$$ by using the substitutions $$u=g(x)$$ and $$du=g’(x)dx$$.
• Example Find $$\int 4(3+4x)^2\,dx$$.
• Example Find $$\int 3u^2\sin(u^3)\,du$$.
• Example Find $$\int \frac{x}{x^2+1}\,dx$$.
• Example Find $$\int \frac{4\sinh(\ln t)}{t}\,dt$$.

### 2.1.2 Substitution in Definite Integrals

• When dealing with definite integrals, you may either convert the boundaries to $$u$$-values, or you must substitute back for the original variable before plugging in boundaries.
• Example Compute $$\int_{1/4}^{1/2} 4(3+4x)^2\,dx$$.
• Example Compute $$\int_0^1 z\sqrt{1-z}\,dz$$.
• Example Compute $$\int_0^{\pi/4}\tan^2\theta\sec^2\theta\,d\theta$$.

### 2.1.3 Antiderivatives of Trigonometric Functions

• The antiderivatives of the basic trig functions (besides sine/cosine) may be derived by using Substitution.
• Example Use Substitution to find $$\int\tan\theta\,d\theta$$.
• Example Prove that $$\int\csc x\,dx = -\ln|\csc x+\cot x|+C$$.

### Exercises for 2.1

1. Find $$\int 3(3x-5)^3\,dx$$.
2. Find $$\int 4e^{r-7}\,dr$$.
3. Find $$\int 4v\sech^2(2v^2+1)\,dv$$.
4. Find $$\int \frac{2e^x}{e^x+3}\,dx$$.
5. Find $$\int 2t^3\sqrt{t^2+1}\,dt$$. (Hint: $$2t^3=2t\cdot t^2$$.)
6. Find $$\int \frac{2(\ln s)^3}{s}\,ds$$.
7. Find $$\int \frac{3\sqrt{x}}{2(x^{3/2}+2)^2}\,dx$$.
8. Find $$\int \frac{\cos(1/y)}{y^2}\,dy$$.
9. Compute $$\int_0^{\pi/12} \sec(3\theta)\tan(3\theta)\,d\theta$$.
10. Compute $$\int_1^2 (6x+3)(x^2+x)^2\,dx$$.
11. Compute $$\int_{\ln 3}^{\ln 8}e^z\sqrt{1+e^z}\,dz$$.
12. Compute $$\int_e^{e^2}\frac{1}{x\ln x}\,dx$$.
13. Use Substitution to find $$\int\cot\theta\,d\theta$$.
14. Multiply by $$\frac{\sec x+\tan x}{\sec x+\tan x}$$ and use Substitution to prove $$\int\sec x\,dx=\ln|\sec x+\tan x|+C$$.
15. (Quiz) Find $$\int 3t^5(t^3+3)^2\,dt$$.
• $$\frac{1}{4}t^4-4t^3+C$$
• $$\frac{1}{4}(t^3+3)^4-(t^3+3)^3+C$$
• $$\frac{1}{2}(t^3+3)^2+4(t^3+3)^3+C$$
16. (Quiz) Evaluate $$\int_0^1 x^2e^{2x^3}\,dx$$.
• $$\frac{1}{6}e^2-\frac{1}{6}$$
• $$\frac{1}{4}e^2-\frac{1}{4}e$$
• $$\frac{1}{3}e-\frac{1}{3}$$

Solutions 1-8

Solutions 9-16

## 2.2 Integration by Parts

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 8.1

### 2.2.1 Parts and the Product Rule

• We may reorder the Product Rule $$\frac{d}{dx}[f(x)g(x)]=g(x)f’(x)+f(x)g’(x)$$ as follows: $$f(x)g’(x)=\frac{d}{dx}[f(x)g(x)]-g(x)f’(x)$$.
• Integrating both sides yields the rule of Integration by Parts: $$\int f(x)g’(x)\,dx=f(x)g(x)-\int g(x)f’(x)\,dx$$.
• This is often abbreviated as $$\int u\,dv=uv-\int v\,du$$ by using the substitutions $$u=f(x)$$ $$du=f’(x)dx$$, $$v=g(x)$$ $$dv=g’(x)dx$$.
• Example Find $$\int 2x\cos(x)\,dx$$.
• Example Find $$\int te^t\,dt$$.
• Occasionally you’ll need to use parts twice.
• Example Find $$\int 3x^2\sinh(x)\,dx$$.
• Especially tricky problems may involve cycling back to the original integral.
• Example Find $$\int e^w\sin(2w)\,dw$$.

### 2.2.2 Integrating Definite Integrals by Parts

• When using parts to evaluate definite integrals, do not forget to apply the bounds of integration to the entire integral.
• Example Find $$\int_0^1 s^2e^s\,ds$$.

### 2.2.3 Antiderivatives of Logarithms

• Integrating logarithms is based on integration by parts.
• Example Use Integration by Parts to find $$\int\ln x\,dx$$.

### Exercises for 2.2

1. Find $$\int 3x\cosh(x)\,dx$$.
2. Find $$\int te^{2t}\,dt$$.
3. Find $$\int y^2\sin(y)\,dy$$.
4. Find $$\int 4x\sec^2(x)\,dx$$. (Hint: recall $$\int\tan\theta\,d\theta=\ln|\sec\theta|+C$$.)
5. Find $$\int e^{3w}\sinh(w)\,dw$$.
6. Find $$\int \sin(2x)\cos(4x)\,dx$$.
7. Compute $$\int_1^e x\ln x\,dx$$.
8. (Optional) Find $$\int x^4e^x\,dx$$.
9. (Optional) Prove $$\int \cos^{n+2} x\,dx = \frac{\cos^{n+1} x\sin x}{n+2}+\frac{n+1}{n+2}\int\cos^n x\,dx$$. (Hint: take the derivative of both sides.)
10. (Optional) Find $$\int \cos^4 x\,dx$$ using the above formula.
11. (Quiz) Find $$\int x\cosh x\,dx$$.
• $$x\sinh x-\cosh x+C$$
• $$x^2\sinh x+2\cosh x+C$$
• $$x\cosh x+3x\sinh x+C$$
12. (Quiz) Find $$\int e^\theta\sin\theta\,d\theta$$.
• $$\frac{e^\theta\sin\theta+e^\theta\cos\theta}{3}+C$$
• $$-\frac{e^\theta\cos\theta}{4}+C$$
• $$\frac{e^\theta\sin\theta-e^\theta\cos\theta}{2}+C$$

Solutions 1-5

Solutions 6-12

## 2.3 Trigonometric Integrals

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 8.2

### 2.3.1 Integrating Products of Sine and Cosine

• To integrate a function of the form $$\sin^m x\cos^n x$$ where at least one of $$m,n$$ is odd, use these identities to substitute $$u=\sin x,du=\cos x\,dx$$ or $$u=\cos x,du=-\sin x\,dx$$.
• $$\cos^2 x=1-\sin^2 x$$
• $$\sin^2 x=1-\cos^2 x$$
• Example Find $$\int\sin^3\theta\cos^4\theta\,d\theta$$.
• Example Find $$\int\sin^2(2y)\cos^5(2y)\,dy$$.
• If both $$m,n$$ are even, then one of these identities must be used:
• $$\cos^2 x=\frac{1}{2}+\frac{1}{2}\cos(2x)$$
• $$\sin^2 x=\frac{1}{2}-\frac{1}{2}\cos(2x)$$
• Example Find $$\int\cos^2 x\,dx$$.
• Example Find $$\int\sin^2 z\cos^2 z\,dz$$.

### 2.3.2 Integrating Products of Secant and Tangent

• To integrate a function of the form $$\sec^m x\tan^n x$$, use these identities to substitute $$u=\tan x,du=\sec^2 x\,dx$$ or $$u=\sec x,du=\sec x\tan x\,dx$$.
• $$\tan^2 x=\sec^2 x-1$$
• $$\sec^2 x=\tan^2 x+1$$
• Example Find $$\int\tan^3\theta\sec^3\theta\,d\theta$$.
• Example Find $$\int\sec^4 x\tan^5 x\,dx$$.

### Exercises for 2.3

1. Find $$\int\sin^4 x\cos^3 x\,dx$$.
2. Find $$\int\sin^5 \theta\cos^2 \theta\,d\theta$$.
3. Find $$\int\sin^2 x\,dx$$.
4. Find $$\int\cos^4 y\,dy$$.
5. Find $$\int\tan^2 t\sec^4 t\,dt$$.
6. (Optional) Use integration by parts with cycling to prove $$\int\sec^3 x\,dx = \frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C$$. (Hint: $$\int\sec x\tan^2 x\,dx=\int\sec x(\sec^2 x-1)\,dx$$.)

Solutions

## 2.4 Trigonometric Substitution

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 8.3

### 2.4.1 Substituting for $$a+bx^2$$

• To eliminate factors of the form $$a+bx^2$$ from an integral, use the substitution $$a+bx^2=a+a\tan^2\theta=a\sec^2\theta$$ with $$-\pi/2<\theta<\pi/2$$.
• Example Find $$\int\frac{z^2}{4+9z^2}\,dz$$.
• Example Compute $$\int_0^2\frac{1}{\sqrt{16+4x^2}}\,dx$$. (Recall $$\int\sec\theta\,d\theta=\ln|\sec\theta+\tan\theta|+C$$.)

### 2.4.2 Substituting for $$a-bx^2$$

• To eliminate factors of the form $$a-bx^2$$ from an integral, use the substitution $$a-bx^2=a-a\sin^2\theta=a\cos^2\theta$$ with $$-\pi/2\leq\theta\leq\pi/2$$.
• Note that this is only valid when $$|x|\leq\sqrt{a/b}$$, which is guaranteed when $$a-bx^2$$ is under a square root.
• Example Find $$\int(4-25s^2)^{-3/2}\,ds$$.
• Example Find $$\int\frac{x^3}{\sqrt{1-4x^2}}\,dx$$.

### 2.4.3 Substituting for $$bx^2-a$$

• To eliminate factors of the form $$bx^2-a$$ from an integral, use the substitution $$bx^2-a=a\sec^2\theta-a=a\tan^2\theta$$ with $$0\leq\theta<\pi/2$$.
• Note that this is only valid when $$x\geq\sqrt{a/b}$$, which will be assumed in our problems.
• Example Prove $$\int\frac{1}{x\sqrt{x^2-1}}\,dx=\inverse\sec x+C$$ where $$x>1$$.
• Example Find $$\int\frac{\sqrt{y^2-16}}{y}\,dy$$ where $$y\geq 4$$.

### 2.4.4 Using Inverse Trigonometric Antiderivatives

• Sometimes, a simpler substiution may be combined with the following antiderivatives to obtain a solution more elegantly.
• $$\int\frac{1}{1+x^2}\,dx=\inverse\tan x+C$$.
• $$\int\frac{1}{\sqrt{1-x^2}}\,dx=\inverse\sin x + C$$.
• $$\int\frac{1}{x\sqrt{x^2-1}}\,dx=\inverse\sec x+C$$ where $$x>1$$.
• Example Find $$\int\frac{3}{\sqrt{9-x^2}}\,dx$$ without using a trigonometric substitution.

### Exercises for 2.4

1. Find $$\int\frac{2}{\sqrt{1+4z^2}}\,dz$$. (Recall $$\int\sec\theta\,d\theta=\ln|\sec\theta+\tan\theta|+C$$.)
2. Find $$\int\frac{x^3}{9+x^2}\,dx$$. (Recall $$\int\tan\theta\,d\theta=\ln|\sec\theta|+C$$.)
3. Find $$\int \frac{4}{(1-y^2)^{3/2}}\,dy$$.
4. Find $$\int\frac{2x^3}{\sqrt{9-x^2}}\,dx$$.
5. Prove $$\int\frac{1}{\sqrt{1-x^2}}\,dx=\inverse\sin x+C$$.
6. Find $$\int\frac{\sqrt{x^2-16}}{x}\,dx$$ where $$x\geq 4$$.
7. Find $$\int\frac{1}{\sqrt{4t^2-1}}\,dt$$ where $$t>\frac{1}{2}$$.
8. Find $$\int\frac{2}{\sqrt{1-4x^2}}\,dx$$ without a trigonometric substitution.
9. (Optional) Find $$\int\frac{2}{4+9x^2}\,dx$$ without a trigonometric substitution.
10. (Quiz) Find $$\int \frac{1}{\sqrt{9+y^2}}\,dy$$. (Recall $$\int\sec\theta\,d\theta=\ln|\sec\theta+\tan\theta|+C$$.)
• $$\ln|\sqrt{1+\frac{1}{9}y^2}+\frac{y}{3}|+C$$.
• $$\sin^{\leftarrow}(\frac{y}{9})+C$$
• $$\ln(\sqrt{9+\frac{1}{9}y^2})+C$$
• (Note: the quiz given in class had a typo: $$\int \frac{1}{\sqrt{9-y^2}}\,dy$$, making $$\inverse\sin(\frac{y}{3})+C$$ or “None of the Above” the correct solution. The full solution below is for the version without a typo.)
11. (Quiz) Find $$\int \frac{1}{x\sqrt{4x^2-1}}\,dy$$ where $$x>\frac{1}{2}$$.
• $$\tan^{\leftarrow}(4x^2-1)+\ln|x|+C$$
• $$\sec^{\leftarrow}(2x)+C$$
• $$\ln|x+\sqrt{4x^2-1}|+C$$

Solutions 1-5

Solutions 6-11

## 2.5 Integrating with Partial Fractions

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 8.4

### 2.5.1 Rational Functions and Partial Fractions

• A function of the form $$\frac{f(x)}{g(x)}$$ where $$f,g$$ are both polynomials is called rational.
• The rational function $$\frac{f(x)}{(x+r)^m}$$ may be split into the partial fractions $$\frac{A_1}{x+r}+\frac{A_2}{(x+r)^2}+\dots+\frac{A_m}{(x+r)^m}$$, provided the degree of the numerator is less than the denominator.
• Example Expand $$\frac{2x^2-7x+6}{(x-2)^3}$$ using partial fractions.
• The rational function $$\frac{f(x)}{(x^2+px+q)^n}$$ (where $$x^2+px+q$$ is irreducible) may be split into the partial fractions $$\frac{B_1x+C_1}{x^2+px+q}+ \frac{B_2x+C_2}{(x^2+px+q)^2}+ \dots+ \frac{B_mx+C_m}{(x^2+px+q)^n}$$, provided the degree of the numerator is less than the denominator.
• Example Expand $$\frac{3x^2+2x+4}{x^4+2x^2+1}$$ using partial fractions.
• When the denominator is a product of $$(x+r)^m$$ and $$(x^2+px+q)^n$$ terms, simply sum up the appropriate partial fractions for each factor.
• Example Describe the partial fractions which expand the rational function $$\frac{f(x)}{(x+3)^3(x^2-2x+3)^2}$$.

### 2.5.2 Integrating Partial Fractions

• Expanding rational functions using partial fractions allows us to integrate.
• Example Find $$\int\frac{2x^2+5x-9}{(x-1)(x+1)(x-2)}\,dx$$.
• Example Find $$\int\frac{4y^2+14y+15}{y^3+4y^2+5y}\,dy$$.
• If the numerator has degree greater than or equal to the denominator, you will need to use long polynomial division to break down the rational function first.
• Example Find $$\int\frac{2t^3+t^2+3t+2}{(1+t)(1+t^2)}\,dt$$.

### Exercises for 2.5

1. Expand $$\frac{4x^2+16x+17}{(x+2)^3}$$ using partial fractions.
2. Expand $$\frac{-y^2+2y-4}{(y^2+4)^2}$$ using partial fractions.
3. Expand $$\frac{3r^3+r^2+3}{r^4+3r^2}$$ using partial fractions.
4. Find $$\int\frac{3z+2}{z^2+2z+1}\,dz$$.
5. Find $$\int\frac{3x^2+35}{x^3+5x}\,dx$$.
6. Find $$\int\frac{2v^3+4v^2+4v+2}{v^2+2v}\,dv$$.
7. (Optional) Find $$\int\frac{2x^3+6x^2+4x+2}{(x+1)^2(x^2+1)}\,dx$$.
8. (Quiz) Which of the following describes the expansion of $$\frac{f(t)}{(t+1)^2(t^2+9)}$$ using partial fractions? (Assume $$f(t)$$ is a polynomial of degree less than 4.)
• $$\frac{At+B}{t+1}+\frac{C}{t^2+1}+\frac{D}{t^2+9}$$.
• $$\frac{A}{t}+\frac{Bt+C}{(t+1)^2}+\frac{D}{t+3}+\frac{E}{t^2+9}$$
• $$\frac{A}{t+1}+\frac{B}{(t+1)^2}+\frac{Ct+D}{t^2+9}$$
9. (Quiz) Find $$\int \frac{-x^2+6x-3}{(x+3)(x^2+1)}\,dx$$.
• $$-3\ln|x+3|+\ln|x^2+1|+C$$
• $$\frac{3}{x^2+9}+2\ln(x^2+1)+C$$
• $$2\ln(x+3)-\tan^{\leftarrow}(x^2+1)+C$$

Solutions 1-4

Solutions 5-9

## 2.6 Strategies for Integration

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• Review of 5.5, 8.1, 8.2, 8.3, 8.4

### 2.6.1 Identifying Appropriate Integration Strategies

• When encountering an integration problem, it’s useful to spot certain traits which can identify the best integration method to apply. The following list isn’t fool-proof, but checking these in order can help you identify likely techniques for integration.
1. Use algebra to simplify the integrand first, if possible. Split up sums into separate integrals as necessary.
2. Is the integrand a constant multiple of a known derivative? If so, simply integrate using the constant multiple rule.
3. Is the integral of the form $$\int cf(g(x))g’(x)\,dx$$: a nested function along with (a constant multiple of) its derivative? If so, use integration by substitution with $$u=g(x)$$.
4. Is the integrand a rational function (a fraction of two polynomials)? If so, try the method of partial fractions to expand the integrand algebraically.
5. Does the integrand include only trigonometric functions? Use trig identities to allow for a direct substitution.
6. Does the integrand include expressions of the form $$a+bx^2$$, $$a-bx^2$$, or $$bx^2-a$$? Use the method of trigonometric substitution to simplify.
7. Is the integrand the product of two functions? Integration by parts may produce a more manageable integral.
8. At this point, check to make sure you didn’t miss a possibility above. Otherwise, you may need to use a combination of techniques from the above to proceed.
• Example Find $$\int\sinh x\sqrt{1+\cosh x}\,dx$$.
• Example Find $$\int 2ze^{3z}\,dz$$.
• Example Find $$\int\sin^2 \theta+\cos^2 \theta\,d\theta$$.
• Example Find $$\int\frac{5x^2+12}{x^3+4x}\,dx$$.
• Example Find $$\int3\sec y\tan y-\frac{1}{1+y^2}\,dy$$.
• Example Find $$\int\frac{1}{\sqrt{4-9t^2}}\,dt$$.
• Example Find $$\int\sin^2 x\cos^3 x\,dx$$.

### Exercises for 2.6

1. Find $$\int(x^2-1)(x^2+1)\,dx$$.
2. Find $$\int\frac{1}{\sqrt{9+z^2}}\,dz$$. (Recall $$\int\sec\theta\,d\theta=\ln|\sec\theta+\tan\theta|+C$$.)
3. Find $$\int 6y^2e^{y^3}\,dy$$.
4. Find $$\int 3x\sin(4x)\,dx$$.
5. Find $$\int\sec^3 \theta\tan^3 \theta\,d\theta$$.
6. Find $$\int\frac{5x-5}{x^2-3x-4}\,dx$$.
7. Find $$\int \frac{3}{2}\sqrt{t}-\frac{1}{t\sqrt{t^2-1}}\,dt$$.
8. (Optional) Find $$\int e^x\sqrt{1-e^{2x}}\,dx$$. (Hint: $$\sin(2\theta)=2\sin\theta\cos\theta$$.)

Solutions

## Review Exercises

The exercises are now located with their respective notes.