Example 21 - Find integral ex sin x dx - Class 12 NCERT

Example 21 Find ﷮﷮ 𝑒﷮𝑥﷯﷯ sin﷮𝑥﷯ 𝑑𝑥 Let I1 = ﷮﷮ 𝑒﷮𝑥﷯﷯ sin﷮𝑥﷯ 𝑑𝑥 Hence, we take First function :- 𝑓 𝑥﷯= sin﷮𝑥﷯ Second function :- g 𝑥﷯= 𝑒﷮𝑥﷯ I1 = sin﷮𝑥﷯ ﷮﷮ 𝑒﷮𝑥﷯ 𝑑𝑥﷯− ﷮﷮ 𝑑 sin﷮𝑥﷯﷯﷮𝑑𝑥﷯ ﷮﷮ 𝑒﷮𝑥﷯ 𝑑𝑥﷯﷯﷯ 𝑑𝑥 I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− ﷮﷮ cos﷮𝑥﷯ . 𝑒﷮𝑥﷯ 𝑑𝑥﷯ Let I2 = ﷮﷮ cos﷮𝑥﷯ . 𝑒﷮𝑥﷯ 𝑑𝑥﷯ Hence, we take First function :- 𝑓 𝑥﷯= cos﷮𝑥﷯ Second function :- g 𝑥﷯= 𝑒﷮𝑥﷯ I2 = cos x ﷮﷮ 𝑒﷮𝑥﷯ 𝑑𝑥﷯ – ﷮﷮(( cos﷮𝑥﷯)′﷯ ﷮﷮ 𝑒﷮𝑥﷯𝑑𝑥﷯)𝑑𝑥 I2 = cos x 𝑒﷮𝑥﷯ – ﷮﷮(− sin﷮𝑥﷯)﷯ 𝑒﷮𝑥﷯ 𝑑𝑥 I2 = 𝑒﷮𝑥﷯ cos x + ﷮﷮ sin﷮𝑥﷯﷯ 𝑒﷮𝑥﷯ 𝑑𝑥 I2 = 𝑒﷮𝑥﷯ cos x + 𝐼1 Now, Putting value of I2 in (1) , I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− ﷮﷮ cos﷮𝑥﷯ 𝑒﷮𝑥﷯﷯ 𝑑𝑥 I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯+𝐼1﷯ I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯−𝐼1 2I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯ I1 = 1﷮2﷯ 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯ ﷯ + C I1 = 𝑒﷮𝑥﷯﷮2﷯ sin﷮𝑥﷯− cos﷮𝑥﷯ ﷯ + C I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯+ ﷮﷮ 𝑒﷮𝑥﷯﷯ sin﷮𝑥﷯ 𝑑𝑥+𝐶1﷯ I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯+ ﷮﷮ 𝑒﷮𝑥﷯﷯ sin﷮𝑥﷯ 𝑑𝑥−𝐶1 I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯−𝐼1−𝐶1 2𝐼1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯− 𝑒﷮𝑥﷯ cos﷮𝑥﷯−𝐶1 I1 = 𝑒﷮𝑥﷯ sin﷮𝑥﷯ − 𝑒﷮𝑥﷯ cos﷮𝑥﷯﷮2﷯ + 𝐶1 ﷮2﷯ I1 = 𝒆﷮𝒙﷯﷮𝟐﷯( 𝒔𝒊𝒏﷮𝒙﷯ − 𝒄𝒐𝒔﷮𝒙﷯)+𝐂