integrate x*e^x
Let's integrate: <math><mo>∫(</mo><mi>x</mi><mo>·</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>)dx</mo></math> For <math><mi>x</mi><mo>·</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math>, we use <u>integration by parts</u>: <math><mo>∫(</mo><mi>u1</mi><mo>·</mo><mi>v2'</mi><mo>)dx</mo><mo> = </mo><mi>u1</mi><mo>·</mo><mi>v2</mi><mo> + </mo><mo>∫(</mo><mi>u1'</mi><mo>·</mo><mi>v2</mi><mo>)dx</mo></math>, with <m>u1 = <math><mi>x</mi></math></m> and <m>v2' = <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math></m> So the value of <m><math><mi>u1'</mi></math></m> is <math><mn>1</mn></math> Let's compute the integral <m>v2'</m>: <math><mo>∫(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>)dx</mo></math> <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> Let's compute the integral <math><mi>u1</mi><mo>·</mo><mi>v2</mi><mo> + </mo><mo>∫(</mo><mi>u1'</mi><mo>·</mo><mi>v2</mi><mo>)dx</mo></math>: <math><mi>x</mi><mo>·</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo> -</mo><mo>∫(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>)dx</mo></math> <math><mi>x</mi><mo>·</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo> -</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> You can factorize this expression into: <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>·</mo><mo>(</mo><mi>x</mi><mo> -</mo><mn>1</mn><mo>)</mo></math> Indefinite integrals are defined up to an additive constant <m>C</m>, so this yields: <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>·</mo><mo>(</mo><mi>x</mi><mo> -</mo><mn>1</mn><mo>)</mo><mo> + </mo><mi>C</mi></math> ▷<b>Answer: <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>·</mo><mo>(</mo><mi>x</mi><mo> -</mo><mn>1</mn><mo>)</mo><mo> + </mo><mi>C</mi></math></b><span style='font-size:smaller'>, with <m>C</m> being any constant (computation required 9 steps and 41 ms)</span>
