Evaluate the following:
1. [tex]\int \frac{dx}{(x^2-16)^3}[/tex]
2. [tex]\int \frac{dx}{(1-x^2)^2}[/tex]
3. [tex]\int (x^2-1)^{\frac{5}{2}} dx[/tex]
4. [tex]\int (4-x^2)^{\frac{3}{2}}dx[/tex]
5. [tex]\int \frac{\sqrt{9-4x^2}}{x} dx[/tex]
6. [tex]\int \frac{x^2}{\sqrt{x^2-4}} dx[/tex]
7. [tex]\int \frac{x^2}{\sqrt{x^2-4}} dx[/tex]
8. [tex]\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6} dx[/tex]
9. [tex]\int \frac{dx}{x\sqrt{9+4x^2}}[/tex]
10. [tex]\int \frac{dx}{x^2\sqrt{9-4x^2}}[/tex]
Answer (2)
Let's evaluate the given integrals step-by-step:
Evaluate ∫ ( x 2 − 16 ) 3 d x : This integral can be approached using partial fraction decomposition once transformed into a form involving rational functions. It may also require a substitution or recognition of a special integral form.
Evaluate ∫ ( 1 − x 2 ) 2 d x : This integral suggests substitution using x = sin ( θ ) , which helps convert the expression involving trigonometric identities.
Evaluate ∫ ( x 2 − 1 ) 2 5 d x : This requires using a binomial expansion or trigonometric substitution if evaluated without a calculator.
Evaluate ∫ ( 4 − x 2 ) 2 3 d x : Using a trigonometric substitution where x = 2 sin ( θ ) or x = 2 cos ( θ ) simplifies the expression. Transformation into trigonometric forms can make this tractable.
Evaluate ∫ x 9 − 4 x 2 d x : This can be rewritten using trigonometric substitution, such as x = 2 3 sin ( θ ) , and simplified.
Evaluate ∫ x 2 − 4 x 2 d x : Here, a substitution like x = 2 sec ( θ ) can help integrate the expression.
Evaluate ∫ x 2 − 4 x 2 d x : This appears to be a repeat of a previous integral and would hence have the same procedure applied.
Evaluate ∫ x 6 ( 16 − 9 x 2 ) 2 3 d x : First, simplify ( 16 − 9 x 2 ) using trigonometric substitution, then detail partial fractions on the result.
Evaluate ∫ x 9 + 4 x 2 d x : A substitution such as x = 2 3 tan ( θ ) gives a clearer integral.
Evaluate ∫ x 2 9 − 4 x 2 d x : Solve using trigonometric substitutions like x = 2 3 sin ( θ ) to simplify under radicals into trigonometric expressions.
Each of these integrals can be complex, and typically require knowledge of specific integrals or substitutions that transform them into solvable forms. The key here is selecting the appropriate technique such as substitution, partial fractions, or trigonometric identities.
The integrals can be evaluated using techniques like substitution and partial fractions. Each integral may require different substitutions or methods to simplify them effectively. Understanding trigonometric identities and transformations is crucial for these evaluations.
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