Fill in the spots in the integration process below based on the indefinite integral 4x²-25 S using the trigonometric substitution: type (theta) for angle 2x dx then the √4x² – 25 = and da after replacement into integral and simplifying we are left with 5 ftan² (0) de 5 f = de
Fill in the spots in the integration process below based on the indefinite integral 4x²-25 S using the trigonometric substitution: type (theta) for angle 2x dx then the √4x² – 25 = and da after replacement into integral and simplifying we are left with 5 ftan² (0) de 5 f = de
Calculus: Early Transcendentals
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Author:James Stewart
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![### Integration Process with Trigonometric Substitution
**Indefinite Integral to be Solved:**
\[
\int \frac{\sqrt{4x^2 - 25}}{x} \, dx
\]
**Trigonometric Substitution:**
Using the trigonometric identity, substitute with an angle \(\theta\):
1. \( 2x = \) **(Blank for expression involving \(\theta\))**
2. \( dx = \) **(Blank for derivative regarding \(d\theta\))**
**Expression Simplification:**
Calculate:
\[ \sqrt{4x^2 - 25} = \] **(Blank for simplified expression)**
**Integral Simplification:**
After substitution and simplification, the integral becomes:
\[
5 \int \tan^2(\theta) \, d\theta = 5 \int \, \text{(Blank for simplified integral)} \, d\theta
\]
### Explanation:
By using the appropriate trigonometric substitution, the integration of a complex algebraic expression can be simplified. The blanks indicate where specific expressions or derivatives should be filled in to complete the substitution process. Trigonometric identities are key for reducing the radical expression, leading to a simpler integral involving trigonometric functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24fae4ed-b1b1-4171-bba8-4d29349ae5e5%2F5eb72f80-9c1d-438b-b1af-196170b6e2b3%2F0eaq8n9_processed.png&w=3840&q=75)
Transcribed Image Text:### Integration Process with Trigonometric Substitution
**Indefinite Integral to be Solved:**
\[
\int \frac{\sqrt{4x^2 - 25}}{x} \, dx
\]
**Trigonometric Substitution:**
Using the trigonometric identity, substitute with an angle \(\theta\):
1. \( 2x = \) **(Blank for expression involving \(\theta\))**
2. \( dx = \) **(Blank for derivative regarding \(d\theta\))**
**Expression Simplification:**
Calculate:
\[ \sqrt{4x^2 - 25} = \] **(Blank for simplified expression)**
**Integral Simplification:**
After substitution and simplification, the integral becomes:
\[
5 \int \tan^2(\theta) \, d\theta = 5 \int \, \text{(Blank for simplified integral)} \, d\theta
\]
### Explanation:
By using the appropriate trigonometric substitution, the integration of a complex algebraic expression can be simplified. The blanks indicate where specific expressions or derivatives should be filled in to complete the substitution process. Trigonometric identities are key for reducing the radical expression, leading to a simpler integral involving trigonometric functions.
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