NGUYÊN HÀM, TÍCH PHÂN, ỨNG DỤNG
Bài 41 ( SGK trang 175 Giải tích 12 NC)
a) y=2x(1−x−3);y=2x(1−x−3); b) y=8x−2x14;y=8x−2x14;
c) y=x12sin(x32+1);y=x12sin(x32+1); d) y=sin(2x+1)cos2(2x+1);y=sin(2x+1)cos2(2x+1);
Giải
a) ∫2x(1−x−3)dx=∫(2x−2x−2)dx=x2+2x+C∫2x(1−x−3)dx=∫(2x−2x−2)dx=x2+2x+C
b) ∫(8x−2x14)dx=∫(8x−2x−14)dx=4x2−83x34+C∫(8x−2x14)dx=∫(8x−2x−14)dx=4x2−83x34+C
c) Đặt
u=x32+1⇒du=32x12dx⇒x12dx=23du∫x12sin(x32+1)dx=23∫sinudu=−23cosu+C=−23cos(x32+1)+Cu=x32+1⇒du=32x12dx⇒x12dx=23du∫x12sin(x32+1)dx=23∫sinudu=−23cosu+C=−23cos(x32+1)+C
d) Đặt u=cos(2x+1)⇒du=−2sin(2x+1)dx⇒sin(2x+1)dx=−12duu=cos(2x+1)⇒du=−2sin(2x+1)dx⇒sin(2x+1)dx=−12du
Do đó ∫sin(2x+1)cos2(2x+1)dx=−12∫duu2=12u+C=12cos(2x+1)+C
Bài 42 ( SGK trang 175 Giải tích 12 NC)
a) y=1x2cos(1x−1)y=1x2cos(1x−1); b) y=x3(1+x4)3y=x3(1+x4)3;
c) y=xe2x3y=xe2x3; d) y=x2exy=x2ex.
Giải
a) Đặt u=1x−1⇒du=−1x2dx⇒dxx2=−duu=1x−1⇒du=−1x2dx⇒dxx2=−du
Do đó ∫1x2cos(1x−1)dx=−∫cosudu=−sinu+C=−sin(1x−1)+C∫1x2cos(1x−1)dx=−∫cosudu=−sinu+C=−sin(1x−1)+C
b) Đặt u=1+x4⇒du=4x3dx⇒x3dx=du4u=1+x4⇒du=4x3dx⇒x3dx=du4
∫x3(1+x4)3dx=14∫u3du=u416+C=116(1+x4)4+C∫x3(1+x4)3dx=14∫u3du=u416+C=116(1+x4)4+C
c) Đặt
{u=x3dv=e2xdx⇒{du=13dxv=12e2x{u=x3dv=e2xdx⇒{du=13dxv=12e2x
Suy ra: ∫xe2x3dx=16xe2x−16∫e2xdx=16xe2x−112e2x+C∫xe2x3dx=16xe2x−16∫e2xdx=16xe2x−112e2x+C
d) Đặt
{u=x2dv=exdx⇒{du=2xdxv=ex{u=x2dv=exdx⇒{du=2xdxv=ex
Suy ra ∫x2exdx=x2ex−2∫xexdx∫x2exdx=x2ex−2∫xexdx (1)
Đặt
{u=xdv=exdx⇒{du=dxv=ex{u=xdv=exdx⇒{du=dxv=ex
Do đó: ∫xexdx=xex−∫exdx=xex−ex+C∫xexdx=xex−∫exdx=xex−ex+C
Từ (1) suy ra ∫x2exdx=x2ex−2xex+2ex+C=ex(x2−2x+2)+C