# Section 2.1 Calculus 2

Integration by Substitution

## 2.1 Integration by Substitution

### 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

• Example Use Substitution to find $$\int\tan\theta\,d\theta$$.
• The antiderivatives of the basic trig functions may be found by using Substitution.
• $$\int\cos x\,dx = \sin x+C$$.
• $$\int\sin x\,dx = -\cos x+C$$.
• $$\int\sec x\,dx = \ln|\sec x+\tan x|+C$$.
• $$\int\csc x\,dx = -\ln|\csc x+\cot x|+C$$.
• $$\int\tan x\,dx = \ln|\sec x|+C$$.
• $$\int\cot x\,dx = -\ln|\csc x|+C$$.
• Example Prove that $$\int\csc x\,dx = -\ln|\csc x+\cot x|+C$$.

### Review Exercises

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. Find $$\int 3t^5(t^3+3)^2\,dt$$.
16. Evaluate $$\int_0^1 x^2e^{2x^3}\,dx$$.

Solutions 1-8

Solutions 9-16

#### Textbook References

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