The use of integration by parts transforms ∫ab x cos (4 x) d x to

Question

The use of integration by parts transforms ∫ab x cos (4 x) d x to (A) (b/4) sin (4 b)-(a/4) sin (4 a)-(1/4) ∫ab sin (4 x) d x (B) ∫ab x d x+∫ab cos (4 x) d x (C) b sin (4 b)-a sin (4 a)-∫ab sin (4 x) d x (D) ( b /2) sin (4 b )-( a /2) sin (4 a )+(1/4) ∫ a b sin (4 x ) dx

Tardigrade Admin · Accepted Answer

(x/4) sin 4 x|a b-(1/4) ∫ limitsab sin (4 x) d x -(a/4) sin (4 a)+(b/4) sin (4 b)-(1/4) ∫ limitsab sin (4 x) d x ⇒ text (A)

Q. The use of integration by parts transforms $\int_{a}^{b} x cos (4 x) d x$ to

$\frac{b}{4} sin (4 b) - \frac{a}{4} sin (4 a) - \frac{1}{4} \int_{a}^{b} sin (4 x) d x$

$\int_{a}^{b} x d x + \int_{a}^{b} cos (4 x) d x$

$b sin (4 b) - a sin (4 a) - \int_{a}^{b} sin (4 x) d x$

$\frac{b}{2} sin (4 b) - \frac{a}{2} sin (4 a) + \frac{1}{4} \int_{a}^{b} sin (4 x) d x$

$\frac{x}{4} sin 4 x ∣_{a}^{b} - \frac{1}{4} a \int b sin (4 x) d x$
$- \frac{a}{4} sin (4 a) + \frac{b}{4} sin (4 b) - \frac{1}{4} a \int b sin (4 x) d x \Rightarrow (A)$

Q. The use of integration by parts transforms ∫ab​xcos(4x)dx to

4b​sin(4b)−4a​sin(4a)−41​∫ab​sin(4x)dx

∫ab​xdx+∫ab​cos(4x)dx

bsin(4b)−asin(4a)−∫ab​sin(4x)dx

2b​sin(4b)−2a​sin(4a)+41​∫ab​sin(4x)dx