Skip to main content

Section Solutions

  1. Integrate by substitution.

    1. \(\dsp 2/9\, \left( {x}^{3}-3 \right) ^{3/2}\)

    2. \(\dsp 2/5(\sqrt {x} +\pi)^5\)

    3. \(\dsp 2/15\, \left( x+5 \right) ^{3/2} \left( -10+3\,x \right)\)

    4. \(\dsp -{\frac {37}{210}}\,\cos \left( 42\,{x}^{5}-6 \right)\)

    5. \(\dsp 1/4\, \left( \sin \left( {x}^{2} \right) \right) ^{2}\text{,}\) \(-\dsp 1/4\, \left( \cos \left( {x}^{2} \right) \right) ^{2}\)

  2. Integrate by parts.

    1. \(\dsp -( 1+x ) {e}^{-x}\)

    2. \(\dsp 1/4\,\sin \left( 2\,x \right) -1/2\,x\cos \left( 2\,x \right)\)

    3. \(\dsp {x}^{2}\cosh \left( x \right) -2\,x\sinh \left( x \right) +2\,\cosh \left( x \right)\)

  3. Integrate by partial fractions.

    1. \(\dsp -\ln |x-1| + \ln |x+1|\)

    2. \(\dsp 2\,\ln \left( x-1 \right) -\ln \left( x+3 \right) +3\,\ln \left( x \right)\)

    3. \(\dsp 5\, \left( x-1 \right) ^{-1}+6\,\ln \left( x-1 \right)\)

    4. \(\dsp \frac{2}{3} \ln \left( 3\,x-5 \right) -5\,{x}^{-1}-7\,\ln \left( x \right)\)

  4. Integrate by trigonometric identities. The answers are computer generated, so may not be in simplest form.

    1. \(\dsp -\cos(x) + \frac{1}{3}\cos^3(x)\)

    2. \(\dsp \frac{1}{4} \cos^3(x) \sin(x) + \frac{3}{8}\cos(x) \sin(x) + \frac{3}{8}x\)

    3. \(\dsp -\frac{1}{9} \sin^9(x) + \frac{1}{7} \sin^7(x)\)

    4. \(\dsp \frac{1}{7} \sec^7(x) - \frac{1}{5} \sec^5(x)\)

    5. \(\dsp -\frac{1}{4} \cos^3(x) \sin(x) +\frac{1}{8} \cos(x) \sin(x) + \frac{1}{8}x\)

  5. Integrate by trigonometric substitution.

    1. \(\dsp \sqrt{x^2-9}-3 \arctan \big( \frac{\sqrt{x^2-9}}{3} \big)\)

    2. \(\dsp \frac{1}{2}\arctan \left( x/2 \right)\)

    3. \(\dsp -x-\ln \left( x-2 \right) +\ln \left( x+2 \right)\)

    4. \(\dsp {\frac {1}{50}}\,{\frac {\sqrt {{x}^{2}-25}}{{x}^{2}}}-{\frac {1}{250} }\,\arctan \left( 5\,{\frac {1}{\sqrt{{x}^{2}-25}}} \right)\)

  6. Integrate these Improper Integrals

    1. diverges

    2. \(\dsp \frac{7\pi}{12}\)

    3. \(\dsp \frac{\pi}{4}\)

    4. diverges

    5. diverges

    6. diverges

    7. the integrand is undefined at \(\pm R\)

  7. Evaluate the indefinite integrals using any method.

    1. \(\dsp 2/3\, \left( x-7 \right) ^{3/2}\text{,}\) \(2/15\, \left( x-7 \right) ^{3/2}\text{,}\) \(\left( 14+3\,x \right) {\frac {2}{105}}\, \left( x-7 \right) ^{3/2} \left( 392+84\,x+15\,{x}^ {2} \right)\)

    2. \(\dsp -\frac{8}{3}\sqrt{8-t^3}\)

    3. \(\dsp -3\, \left( \sin \left( t \right) \right) ^{-1}-2 \cot(t)\)

    4. \(\dsp 2 \tan(\sqrt{x+5})\)

    5. \(2\)

    6. \(\dsp {\frac {\sin \left( ax+b \right) }{a}}\)

    7. \(\dsp 1/4\,{\frac {{3}^{4\,x}}{\ln \left( 3 \right) }}\)

    8. \(\dsp {\frac {{b}^{kx}}{k\ln \left( b \right) }}\)

    9. \(\dsp 1/2 \tan(e^{2x}) + \ln(tan(e^{2x})) - 1/2 \cot(e^{2x})\)

    10. \(\dsp -(2 e^{-x}+e^{2x}/2)\)

    11. \(\dsp e^x + \ln(e^x-5)\)

    12. \(\dsp-2/9\, \left( 4+\sqrt {x} \right) ^{-3}\)

    13. \(\dsp \frac{1}{4}\)

    14. \(\dsp 1/20\, \left( \ln \left( 4\,x \right) \right) ^{4}\)

    15. \(\dsp 1/15\, \left( 2\,x+1 \right) ^{3/2} \left( -1+3\,x \right)\)

  8. No more solutions! Either use an integrator on the web or use a symbolic calculator.

  9. No solutions to hyperbolic problems because all are to verify something is true so you know if you got the right answer!

login