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∫33(y2+3)dy=\int_3^3\left(y^2+3\right)dy=∫33(y2+3)dy=
−∫03y2dy-\int_0^3y^2dy−∫03y2dy
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1
∫13g(x)dx=\int_1^3g\left(x\right)dx=∫13g(x)dx=
∫13g(4+x)dx\int_1^3g\left(4+x\right)dx∫13g(4+x)dx
∫cdg(c+d−v)dv\int_c^dg\left(c+d-v\right)dv∫cdg(c+d−v)dv
∫13g(4−x)dx=\int_1^3g\left(4-x\right)dx=∫13g(4−x)dx=
∫13ydy +∫35ydy =\int_1^3ydy\ +\int_3^5ydy\ =∫13ydy +∫35ydy =
∫31ydy\int_3^1y^{ }dy∫31ydy
∫15ydy\int_1^5y^{ }dy∫15ydy
∫−11y2dy=\int_{-1}^1y^2dy=∫−11y2dy=
2∫01v2dv2\int_0^1v^2dv2∫01v2dv
∫10vdv\int_1^0vdv∫10vdv
∫01v2dv\int_0^1v^2dv∫01v2dv
∫cd5y2dy=\int_c^d5y^2dy=∫cd5y2dy=
∫cdvdv\int_c^dvdv∫cdvdv
5∫cdv2dv=\int_c^dv^2dv=∫cdv2dv=
25∫cdv2dv\int_c^dv^2dv∫cdv2dv
If g is an odd function,
∫−22y3dy=\int_{-2}^2y^3dy=∫−22y3dy=
2∫02v3dv2\int_0^2v^3dv2∫02v3dv
Fill in the blanks:
∫14f(t)dt=\int_1^4f\left(t\right)dt=∫14f(t)dt=
∫41f(v)dv\int_4^1f\left(v\right)dv∫41f(v)dv
∫11f(v)dv=\int_1^1f\left(v\right)dv=∫11f(v)dv=
∫14f(v)dv=\int_1^4f\left(v\right)dv=∫14f(v)dv=
∫14[(5x)−(x2)]dx\int_1^4\left[\left(5x\right)-\left(x^2\right)\right]dx∫14[(5x)−(x2)]dx =
∫145xdx−∫14x2dx\int_1^45xdx-\int_1^4x^2dx∫145xdx−∫14x2dx
∫145xdx\int_1^45xdx∫145xdx
∫145x dx+∫14x2dx\int_1^45x\ dx+\int_1^4x^2dx∫145x dx+∫14x2dx
∫14(4+v)dv=\int_1^4\left(4+v\right)dv=∫14(4+v)dv=
−∫11(4+v)dv-\int_1^1\left(4+v\right)dv−∫11(4+v)dv
∫12(4+v)dv=\int_1^2\left(4+v\right)dv=∫12(4+v)dv=
−∫21(4+v)dv-\int_2^1\left(4+v\right)dv−∫21(4+v)dv
∫03f(x)dx\int_0^3f\left(x\right)dx∫03f(x)dx
2∫30f(v)dv2\int_3^0f\left(v\right)dv2∫30f(v)dv
∫03f(3−v)dv\int_0^3f\left(3-v\right)dv∫03f(3−v)dv
4∫03f(v)dv4\int_0^3f\left(v\right)dv4∫03f(v)dv
It is done.