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

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

Popular >

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

\int_{- 1}^{0} (x + 1) cos (nπ x) d x

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

Solution steps

\int_{- 1}^{0} (x + 1) cos (nπ x) d x

Expand

(x + 1) cos (nπ x) : x cos (πn x) + cos (πn x)

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

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

Simplify

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

\int_{1}^{3} (x^{2} - x) d x

\int_{0}^{π} x \cdot cos (x^{2}) d x

\int_{7}^{49} \frac{1}{x ( ln ( x ) ) ^{4}} d x

\int_{- \frac{1}{8}}^{\frac{1}{8}} (sin (2 π x)) d x

\int_{- 6}^{1} - x^{2} - 3 x d x

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