What is the integral from-1 to 0 of (x+4)cos(npix) ?

The integral from-1 to 0 of (x+4)cos(npix) is (1-(-1)^n)/(pi^2n^2)

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integral from-1 to 0 of (x+4)cos(npix)

\int_{- 1}^{0} (x + 4) cos (nπ x) d x

\frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}}

Solution steps

\int_{- 1}^{0} (x + 4) cos (nπ x) d x

Expand

(x + 4) cos (nπ x) : x cos (πn x) + 4 cos (πn x)

= \int_{- 1}^{0} x cos (πn x) + 4 cos (πn x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{- 1}^{0} x cos (πn x) d x + \int_{- 1}^{0} 4 cos (πn x) d x

\int_{- 1}^{0} x cos (πn x) d x = \frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}}

\int_{- 1}^{0} 4 cos (πn x) d x = 0

= \frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}} + 0

Simplify

= \frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}}

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What is the integral from-1 to 0 of (x+4)cos(npix) ?
The integral from-1 to 0 of (x+4)cos(npix) is (1-(-1)^n)/(pi^2n^2)