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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=x5 and let dv(x)=cos(x).
Then du(x)=5x4.
To find v(x):
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The integral of cosine is sine:
∫cos(x)dx=sin(x)
Now evaluate the sub-integral.
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=5x4 and let dv(x)=sin(x).
Then du(x)=20x3.
To find v(x):
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The integral of sine is negative cosine:
∫sin(x)dx=−cos(x)
Now evaluate the sub-integral.
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=−20x3 and let dv(x)=cos(x).
Then du(x)=−60x2.
To find v(x):
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The integral of cosine is sine:
∫cos(x)dx=sin(x)
Now evaluate the sub-integral.
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=−60x2 and let dv(x)=sin(x).
Then du(x)=−120x.
To find v(x):
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The integral of sine is negative cosine:
∫sin(x)dx=−cos(x)
Now evaluate the sub-integral.
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=120x and let dv(x)=cos(x).
Then du(x)=120.
To find v(x):
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The integral of cosine is sine:
∫cos(x)dx=sin(x)
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
∫120sin(x)dx=120∫sin(x)dx
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The integral of sine is negative cosine:
∫sin(x)dx=−cos(x)
So, the result is: −120cos(x)
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Add the constant of integration:
x5sin(x)+5x4cos(x)−20x3sin(x)−60x2cos(x)+120xsin(x)+120cos(x)+constant