integrate cos(2x)
Let's integrate: <math><mo>∫(</mo><mi>cos</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo>)dx</mo></math> For the <u>math expression</u>: <math><mi>cos</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math>, we know the derivative formula: <math><mo>(</mo><mi>sin</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>)'</mo><mo> = </mo><mi>cos</mi><mo>(</mo><mi>a</mi><mo>)</mo></math>, so we try differentiating with <m>sin</m> Let's differentiate: <math><mo>(</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo>)'</mo></math> Apply the differentiation rule: <m><math><mo>(</mo><mi>sin</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>)'</mo><mo> = </mo><mo>(</mo><mi>f</mi><mo>)'</mo><mo>·</mo><mi>cos</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>cos</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math> <math><mn>2</mn><mo>·</mo><mi>cos</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math> You observe that the computed derivative and the original <u>math expression</u> are the same, except for a coefficient. So the integral is: <math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>·</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></math> <math><mfrac><mrow><mi>sin</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math> Indefinite integrals are defined up to an additive constant <m>C</m>, so this yields: <math><mfrac><mrow><mi>sin</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo> + </mo><mi>C</mi></math> ▷<b>Answer: <math><mfrac><mrow><mi>sin</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo> + </mo><mi>C</mi></math></b><span style='font-size:smaller'>, with <m>C</m> being any constant (computation took 10 steps and 3 ms)</span>
