integre 2*e^(2*x)
Let's compute the integral: <math><mo>∫(</mo><mn>2</mn><msup><mrow><mi>e</mi></mrow><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></msup><mo>)dx</mo></math> We attempt to reverse an integration: starting from <math><mn>2</mn><msup><mrow><mi>e</mi></mrow><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></msup></math> and <math><mo>(</mo><mi>exp</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>)'</mo><mo> = </mo><mi>exp</mi><mo>(</mo><mi>a</mi><mo>)</mo></math>, we differentiate using <m>exp</m> Let's differentiate: <math><mo>(</mo><mi>exp</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo>)'</mo></math> Apply the differentiation rule: <m><math><mo>(</mo><mi>exp</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>)'</mo><mo> = </mo><mo>(</mo><mi>f</mi><mo>)'</mo><mo>·</mo><mi>exp</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></m> with <m>f</m> = <m><math><mn>2</mn><mi>x</mi></math></m> <math><mo>(</mo><mn>2</mn><mi>x</mi><mo>)'</mo><mo>·</mo><mi>exp</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math> <math><mn>2</mn><mi>exp</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math> The computed derivative matches the original <u>math expression</u> up to a constant factor. Therefore, the integral is: <math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>·</mo><mi>exp</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math> <math><mi>exp</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math> Indefinite integrals are defined up to an additive constant <m>C</m>, so this yields: <math><mi>exp</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo> + </mo><mi>C</mi></math> ▷<b>Result: <math><mi>exp</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo> + </mo><mi>C</mi></math></b><span style='font-size:smaller'>, where <m>C</m> is a constant (computation took 10 steps and 2 ms)</span>
