Q 1 TF - Calculus

Question

what is the comparison test for improper integrals?

Step-by-Step Solution

Answer

If $f\left(x\right)\ge g\left(x\right)\ge 0$on the interval $[0,\infty )$ then

1.if${\int }_0^{\infty }f\left(x\right)dx$  converges then so does${\int }_0^{\infty }g\left(x\right)dx$  converges

2.if ${\int }_0^{\infty }g\left(x\right)dx$ diverges then so does${\int }_0^{\infty }f\left(x\right)dx$  diverges

The comparison test for improper integral

If $f\left(x\right)\ge g\left(x\right)\ge 0$on the interval $[0,\infty )$ then

1.if ${\int }_0^{\infty }f\left(x\right) dx$ converges then so does ${\int }_0^{\infty }g\left(x\right)dx$ converges

2.if ${\int }_0^{\infty }g\left(x\right)dx$ diverges then so does ${\int }_0^{\infty }f\left(x\right)dx$ diverges

Step 2: Choose the integration technique

Examine the integrand to determine the best approach: basic rules, substitution, integration by parts, partial fractions, or trigonometric substitution.

Step 3: Perform the integration

Apply the chosen technique step by step, showing all substitutions and intermediate results.

Step 4: Evaluate and simplify

For definite integrals, apply the Fundamental Theorem of Calculus. For indefinite integrals, simplify and include \(+C\).

Step 5: State the final result

Write the final answer clearly.