Evaluate the definite integral by using the method of integration by parts.

Question

Evaluate the definite integral by using the method of integration by parts. ​  ​

Quizplus · Accepted Answer

Using integration by parts with u = x^3 and dv = e^(x^2)dx,
du = 3x^2 dx and v = 1/2 e^(x^2)
∫x^3e^(x^2) dx = (1/2)x^3*e^(x^2) - ∫(1/2)e^(x^2)*3x^2 dx
= (1/2)x^3*e^(x^2) - (3/2)∫x^2 e^(x^2) dx

Using integration by parts again with u = x^2 and dv = e^(x^2)dx,
du = 2x dx and v = 1/2 e^(x^2)
∫x^3e^(x^2) dx = (1/2)x^3*e^(x^2) - (3/2)[(1/2)x^2*e^(x^2) - ∫e^(x^2)*2x dx]
= (1/2)x^3*e^(x^2) - (3/2)(1/2)x^2*e^(x^2) + 3/2∫e^(x^2) dx
= (1/2)x^3*e^(x^2) - (3/2)(1/2)x^2*e^(x^2) + (3/2)e^(x^2) + C
= (1/2)e^(x^2)(x^3 - 3x^2 + 3) + C

Therefore, the answer is option B) ​11eb651f_cb03_88e8_b0a1_c386ee995d1a_TB8255_11.

Evaluate the Definite Integral by Using the Method of Integration