Improper integrals practice problems
Infinite intervals and infinite integrands: replace the bad bound with a limit. 27 problems with worked solutions; in 27 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \int_{1}^{\infty} \frac{1}{\sqrt{x}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{1}^{\infty} \frac{1}{x} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{4} \frac{1}{\sqrt[3]{x}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{\infty} e^{- 4 x} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{\infty} e^{- 2 x} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{4} \frac{1}{x^{\frac{2}{3}}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{\infty} e^{- x} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{1}^{\infty} \frac{1}{x^{\frac{5}{2}}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{\infty} \frac{1}{x^{2} + 9} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{1}^{\infty} \frac{1}{x^{4}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{\infty} \frac{1}{x^{2} + 1} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{2}^{\infty} \frac{1}{x^{\frac{3}{2}}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{8} \frac{1}{\sqrt[3]{x}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{2}^{\infty} \frac{1}{x^{3}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{8} \frac{1}{\sqrt{x}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{\infty} e^{- 3 x} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{2}^{\infty} \frac{1}{x^{2}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{\infty} \frac{1}{x^{2} + 4} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{1} \frac{1}{x^{\frac{2}{3}}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{4} \frac{1}{\sqrt{x}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{1} \frac{1}{\sqrt[3]{x}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{1}^{\infty} \frac{1}{x^{3}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{2}^{\infty} \frac{1}{x^{\frac{5}{2}}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{1}^{\infty} \frac{1}{x^{\frac{3}{2}}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{8} \frac{1}{x^{\frac{2}{3}}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{2}^{\infty} \frac{1}{x^{4}} \, dx \), or show that it diverges.
Evaluate \( \displaystyle \int_{0}^{1} \frac{1}{\sqrt{x}} \, dx \), or show that it diverges.