Question

The use of integration by parts transforms $\int_a^b x \cos (4 x) d x$ to

• A. $\frac{b}{4} \sin (4 b)-\frac{a}{4} \sin (4 a)-\frac{1}{4} \int_a^b \sin (4 x) d x$
• B. $\int_a^b x d x+\int_a^b \cos (4 x) d x$
• C. $b \sin (4 b)-a \sin (4 a)-\int_a^b \sin (4 x) d x$
• D. $\frac{ b }{2} \sin (4 b )-\frac{ a }{2} \sin (4 a )+\frac{1}{4} \int_{ a }^{ b } \sin (4 x ) dx$

Answer 1

Answer: (A)

$\frac{x}{4} \sin 4 x|_a ^b-\frac{1}{4} \int\limits_a^b \sin (4 x) d x$
$-\frac{a}{4} \sin (4 a)+\frac{b}{4} \sin (4 b)-\frac{1}{4} \int\limits_a^b \sin (4 x) d x \Rightarrow \text { (A) }$