# Section 2.3 Calculus 2

Trigonometric Substitution

## 2.3 Trigonometric Substitution

### 2.3.1 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 $$\int\sec\theta\,d\theta=\ln|\sec\theta+\tan\theta|+C$$.)

### 2.3.2 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 $$\int\frac{x^3}{\sqrt{1-4x^2}}\,dx$$.

### 2.3.3 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 $$y\geq 4$$.

### 2.3.4 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.

### Review Exercises

1. Find $$\int\frac{2}{\sqrt{1+4z^2}}\,dz$$.
2. Find $$\int\frac{x^3}{9+x^2}\,dx$$.
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. Find $$\int\frac{2}{4+9x^2}\,dx$$ without a trigonometric substitution.
10. Find $$\int \frac{1}{\sqrt{9+y^2}}\,dy$$.
11. Find $$\int \frac{1}{x\sqrt{4x^2-1}}\,dy$$ where $$x>\frac{1}{2}$$.

Solutions 1-5

Solutions 6-11

#### Textbook References

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