Use Table of Integrals to evaluate the integral. x^{3} \sin(x^{2}

Joan Thompson

Joan Thompson

Answered question

2021-12-29

Use Table of Integrals to evaluate the integral. x3sin(x2+10)dx.

Answer & Explanation

Cheryl King

Cheryl King

Beginner2021-12-30Added 36 answers

x3sin(x2+10)dx
Let x2+10=t
2xdx=dt
12(t10)sintdt
=12[tsintdt10sintdt]
=12[(sinttcost+c)10(cost)]
=12[sinttcost+10cost]+C
=12[sin(x2+10)+10cos(x2+10)(x2+10)cos(x2+10)]+C

hysgubwyri3

hysgubwyri3

Beginner2021-12-31Added 43 answers

x3sin(x2+10) dx  
Let's place the expression 2 * x under the differential's sign, i.e.
2x dx =d(x2),t=x2 
The initial integral can then be expressed as follows: The 
tsin(t+10)2 dt  
xsin(x+10)2 dx  
formula for integration by parts: 
U(x)dV(x)=U(x)V(x)V(x)dU(x) 
Put 
U=x 
dV=sin(x+10)2 dx  
Then: 
dU=dx 
V=cos(x+10)2 
Thus: 
xsin(x+10)2 dx =xcos(x+10)2cos(x+10)2 dx =xcos(x+10)2+cos(x+10)2 dx  
Find the integral 
cos(x+10)2 dx =sin(x+10)2 
Result: 
xsin(x+10)2=xcos(x+10)2+sin(x+10)2+C 
To write down the final answer, it remains to substitute x2 instead of t. 
x2cos(x2+10)2+sin(x2+10)2+C

Vasquez

Vasquez

Expert2022-01-07Added 669 answers

x3×sin(x2+10)dxt×sin(t)10sin(t)2dt12×t×sin(t)10sin(t)dt12×(t×sin(t)dt10sin(t)dt)12×(t×cos(t)+sin(t)+10cos(t))12×((x2+10)×cos(x2+10)+sin(x2+10)+10cos(x2+10))cos(x2+10)×(x210)+sin(x2+10)2+5cos(x2+10)Solution:cos(x2+10)×(x210)+sin(x2+10)2+5cos(x2+10)+C

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