Trigonometric substitution practice problems
Integrals with √(a² − x²), √(a² + x²): substitute x = a sin θ or a tan θ, integrate, and draw the triangle to get back to x. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \int \frac{1}{x^{2} \sqrt{x^{2} + 9}}\, dx \).
Evaluate \( \displaystyle \int \frac{1}{x^{2} \sqrt{25 - x^{2}}}\, dx \).
Evaluate \( \displaystyle \int \sqrt{16 - x^{2}}\, dx \).
Evaluate \( \displaystyle \int \frac{1}{x^{2} \sqrt{x^{2} + 16}}\, dx \).
Evaluate \( \displaystyle \int \sqrt{9 - x^{2}}\, dx \).
Evaluate \( \displaystyle \int \frac{1}{\left(9 - x^{2}\right)^{\frac{3}{2}}}\, dx \).
Evaluate \( \displaystyle \int \frac{x^{2}}{\sqrt{1 - x^{2}}}\, dx \).
Evaluate \( \displaystyle \int \frac{1}{\left(x^{2} + 16\right)^{\frac{3}{2}}}\, dx \).
Evaluate \( \displaystyle \int \frac{1}{x^{2} \sqrt{x^{2} + 4}}\, dx \).
Evaluate \( \displaystyle \int \sqrt{1 - x^{2}}\, dx \).