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Question Number 80432 by Power last updated on 03/Feb/20

Commented by mr W last updated on 03/Feb/20

Commented by john santu last updated on 03/Feb/20

$v e r y s m a l l s i r . h a h a$
Commented by Power last updated on 03/Feb/20

$solution sir$
Commented by mr W last updated on 03/Feb/20

Commented by abdomathmax last updated on 03/Feb/20
$I=∫ e^(3x) sin^4 xdx ⇒I =∫ e^(3x) (((1−cos(2x))/2))^2 dx =(1/4)∫ e^(3x) (cos^2 (2x)−2cos(2x)+1)dx =(1/4)∫ e^(3x) (((1+cos(4x))/2)−2 cos(2x)+1)dx =(1/8)∫ e^(3x) (1 +cos(4x)−4cos(2x)+2)dx =(1/8)∫ e^(3x) (3 +cos(4x)−4cos(2x))dx =(3/8)∫ e^(3x) dx +(1/8)∫ e^(3x) cos(4x)dx−(1/2)∫ e^(3x) cos(2x)dx ∫ e^(3x) dx =(1/3)e^(3x) +c_0 ∫ e^(3x) cos(4x)dx =Re(∫ e^(3x+4ix) dx) ∫ e^((3+4i)x) dx =(1/(3+4i))e^((3+4i)x) +c =((3+4i)/(25))e^(3x) (cos(4x)+isin(4x)) +c =(e^(3x) /(25))(3+4i){cos(4x)+isin(4x)} +c =(e^(3x) /(25)){3cos(4x)+3isin(4x)+4icos(4x)−4sin(4x)} ⇒∫ e^(3x) cos(4x)dx =(e^(3x) /(25))( 3cos(4x)−4sin(4x)) +c_1 ∫ e^(3x) cos(2x)dx =Re(∫ e^((3+2i)x) dx) ∫ e^((3+2i)x) dx =(1/(3+2i))e^((3+2i)x) +c =((3−2i)/(13)) e^(3x) (cos(2x)+isin(2x)) =(e^(3x) /(13)){ 3cos(2x)+3isin(2x)−2icos(2x) +2sin(2x)} ⇒∫ e^(3x) cos(2x)dx =(e^(3x) /(13)){3cos(2x)+2sin(2x)} ⇒ I=(1/8)e^(3x) +(e^(3x) /(8×25)){3cos(4x)−4sin(4x)} −(e^(3x) /(26)){3cos(2x)+2sin(2x)} +C$
$I = \int e^{3 x} {s i n}^{4} x d x \Rightarrow I = \int e^{3 x} {(\frac{1 - c o s (2 x)}{2})}^{2} d x$ $= \frac{1}{4} \int e^{3 x} ({c o s}^{2} (2 x) - 2 c o s (2 x) + 1) d x$ $= \frac{1}{4} \int e^{3 x} (\frac{1 + c o s (4 x)}{2} - 2 c o s (2 x) + 1) d x$ $= \frac{1}{8} \int e^{3 x} (1 + c o s (4 x) - 4 c o s (2 x) + 2) d x$ $= \frac{1}{8} \int e^{3 x} (3 + c o s (4 x) - 4 c o s (2 x)) d x$ $= \frac{3}{8} \int e^{3 x} d x + \frac{1}{8} \int e^{3 x} c o s (4 x) d x - \frac{1}{2} \int e^{3 x} c o s (2 x) d x$ $\int e^{3 x} d x = \frac{1}{3} e^{3 x} + c_{0}$ $\int e^{3 x} c o s (4 x) d x = R e (\int e^{3 x + 4 i x} d x)$ $\int e^{(3 + 4 i) x} d x = \frac{1}{3 + 4 i} e^{(3 + 4 i) x} + c$ $= \frac{3 + 4 i}{25} e^{3 x} (c o s (4 x) + i s i n (4 x)) + c$ $= \frac{e^{3 x}}{25} (3 + 4 i) {c o s (4 x) + i s i n (4 x)} + c$ $= \frac{e^{3 x}}{25} {3 c o s (4 x) + 3 i s i n (4 x) + 4 i c o s (4 x) - 4 s i n (4 x)}$ $\Rightarrow \int e^{3 x} c o s (4 x) d x$ $= \frac{e^{3 x}}{25} (3 c o s (4 x) - 4 s i n (4 x)) + c_{1}$ $\int e^{3 x} c o s (2 x) d x = R e (\int e^{(3 + 2 i) x} d x)$ $\int e^{(3 + 2 i) x} d x = \frac{1}{3 + 2 i} e^{(3 + 2 i) x} + c$ $= \frac{3 - 2 i}{13} e^{3 x} (c o s (2 x) + i s i n (2 x))$ $= \frac{e^{3 x}}{13} {3 c o s (2 x) + 3 i s i n (2 x) - 2 i c o s (2 x) + 2 s i n (2 x)}$ $\Rightarrow \int e^{3 x} c o s (2 x) d x$ $= \frac{e^{3 x}}{13} {3 c o s (2 x) + 2 s i n (2 x)} \Rightarrow$ $I = \frac{1}{8} e^{3 x} + \frac{e^{3 x}}{8 \times 25} {3 c o s (4 x) - 4 s i n (4 x)}$ $- \frac{e^{3 x}}{26} {3 c o s (2 x) + 2 s i n (2 x)} + C$
Commented by Power last updated on 03/Feb/20

$thanks$
Commented by Tony Lin last updated on 03/Feb/20

${s i n}^{4} x = {({s i n}^{2} x)}^{2} = {(\frac{1 - c o s 2 x}{2})}^{2}$ $= \frac{1 + {c o s}^{2} 2 x - 2 c o s 2 x}{4}$ $= \frac{1 + (\frac{1 + c o s 4 x}{2}) - 2 c o s 2 x}{4}$ $= \frac{3}{8} + \frac{c o s 4 x}{8} - \frac{c o s 2 x}{2}$ $\int e^{3 x} {s i n}^{4} x d x$ $= \frac{3}{8} \int e^{3 x} d x + \frac{1}{8} \int e^{3 x} c o s 4 x d x - \frac{1}{2} \int e^{3 x} c o s 2 x d x$ $\int e^{3 x} c o s 4 x d x$ $= \frac{1}{3} e^{3 x} c o s 4 x + \frac{4}{3} \int e^{3 x} s i n 4 x d x$ $= \frac{1}{3} e^{3 x} c o s 4 x + \frac{4}{3} (\frac{1}{3} e^{3 x} s i n 4 x - \frac{4}{3} \int e^{3 x} c o s 4 x d x)$ $\frac{25}{9} \int e^{3 x} c o s 4 x d x = \frac{1}{3} e^{3 x} c o s 4 x + \frac{4}{9} e^{3 x} s i n 4 x + c_{1}$ $\Rightarrow \int e^{3 x} c o s 4 x d x = \frac{3}{25} e^{3 x} c o s 4 x + \frac{4}{25} e^{3 x} s i n 4 x + c_{2}$ $\int e^{3 x} c o s 2 x d x$ $= \frac{1}{3} e^{3 x} c o s 2 x + \frac{2}{3} \int e^{3 x} s i n 2 x d x$ $= \frac{1}{3} e^{3 x} c o s 2 x + \frac{2}{3} (\frac{1}{3} e^{3 x} s i n 2 x - \frac{2}{3} \int e^{3 x} c o s 2 x d x)$ $\frac{13}{9} \int e^{3 x} c o s 2 x d x = \frac{1}{3} e^{3 x} c o s 2 x + \frac{2}{9} e^{3 x} s i n 2 x + c_{3}$ $\Rightarrow \int e^{3 x} c o s 2 x d x = \frac{3}{13} e^{3 x} c o s 2 x + \frac{2}{13} e^{3 x} s i n 2 x + c_{4}$ $\int e^{3 x} d x = \frac{e^{3 x}}{3} + c_{5}$ $∴ \int e^{3 x} {s i n}^{4} x d x$ $= \frac{1}{8} e^{3 x} + \frac{3}{200} e^{3 x} c o s 4 x + \frac{1}{50} e^{3 x} s i n 4 x -$ $\frac{3}{26} e^{3 x} c o s 2 x - \frac{1}{13} e^{3 x} s i n 2 x + c$
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