7.7 Assess Your Understanding

523

Skill Building

In Problems 1–16, find each integral using the Table of Integrals found at the back of the book.

Question

\(\int e^{2x}\cos x\,dx\)

Question

\(\int e^{5x+1}\sin (2x+3)\,dx\)

Question

\(\int x\sqrt{4x+3}\,dx\)

Question

\(\int \dfrac{dx}{(x^{2}-1)^{3/2}}\)

Question

\(\int (x+1) \sqrt{4x+5}\,dx\)

Question

\(\int \dfrac{dx}{[(2x+3)^{2}-1] ^{3/2}}\)

Question

\(\int \dfrac{dx}{x\sqrt{4x+6}}\)

Question

\(\int \dfrac{dx}{x\sqrt{8+x}}\)

Question

\(\int \dfrac{\sqrt{4x+6}}{x}\,dx\)

Question

\(\int \dfrac{\sqrt{8+x}}{x^{2}}\,dx\)

Question

\(\int x^{3}(\ln x)^2\,dx\)

Question

\(\int x^{3} (\ln x)^{2}\,dx\)

Question

\(\int \sin ^{-1}(2x)\,dx\)

Question

\(\int \tan ^{-1}(-3x)\,dx\)

Question

\(\int _{1}^{2}\dfrac{x^{3}}{\sqrt{3x-x^{2}}}\,dx\)

Question

\(\int _{1}^{e}\dfrac{1}{x^{2}\sqrt{x^{2}+2}}\,dx\)

In Problems 17–32:

  1. Redo Problems 1-16 using a CAS.
  2. Compare the result to the answer obtained using a Table of Integrals.
  3. If the results are different, verify that they are equivalent.

Question

\(\int e^{2x}\cos x\,dx\)

Question

\(\int x\sqrt{4x+3}\,dx\)

Question

\(\int (x+1) \sqrt{4x+5}\,dx\)

Question

\(\int \dfrac{dx}{x\sqrt{4x+6}}\)

Question

\(\int \dfrac{\sqrt{4x+6}}{x}\,dx\)

Question

\(\int x^{3}(\ln x)^2\,dx\)

Question

\(\int \sin ^{-1}(2x)\,dx\)

Question

\(\int _{1}^{2}\dfrac{x^{3}}{\sqrt{3x-x^{2}}}\,dx\)

In Problems 33–38, use a CAS to investigate whether each indefinite integral can be expressed using elementary functions.

Question

\(\int \sqrt{1+x^{3}}\,dx\)

Question

\(\int \sqrt{1+\sin x}\,dx\)

Question

\(\int e^{-x^{2}}dx\)

Question

\(\int \dfrac{\cos x}{x}\,dx\)

Question

\(\int x\tan x\, dx\)

Question

\(\int \sqrt{1+e^{x}}\,dx\)