(1.) ∫xe3x
dx here is the problem
Let u = x Let dv = e3x
du = dx v = (1/3)e3x
uv - ∫v du use the parts formula
= (1/3)xe3x - (1/3)∫e3x
dx make substitutions
= (1/3)xe3x - (1/3)(1/3)e3x
+ C integrate
= (1/3)xe3x - (1/9)e3x + C multiply
(2.) ∫xe-7x
dx here is the problem
Let u = x Let dv = e-7x
du = dx v = (-1/7)e-7x
uv - ∫v du use the parts formula
= (-1/7)xe-7x + (1/7)∫e-7x dx make
substitutions
= (-1/7)xe-7x + (1/7)(-1/7)e-7x
+ C integrate
= (-1/7)xe-7x - (1/49)e-7x
+ C multiply
(3.) ∫x2e-x
dx here is the problem
Let u = x2 Let dv = e-x
du = 2x dx v = -e-x
uv - ∫v du use the
parts formula
= -x2e-x + 2∫xe-x dx
make substitutions
Let u = x Let dv = e-x
du = dx v = -e-x
= -x2e-x + 2[uv - ∫v du] use the parts
formula
= -x2e-x - 2xe-x
+ 2∫e-x dx make
substitutions
= -x2e-x - 2xe-x
- 2e-x + C integrate
(4.) ∫x2ex/4
dx
(5.) ∫x ln x
dx here is the problem
Let u = ln x Let dv = x
du = (1/x)dx v = (1/2)x2
uv - ∫v du use the parts formula
= (1/2)x2 ln x - (1/2)∫x2 * (1/x)dx
make substitutions
= (1/2)x2 ln x - (1/2)∫ x
dx cancel
= (1/2)x2 ln x - (1/2)(1/2)x2 + C integrate
= (1/2)x2 ln x - (1/4)x2 + C multiply
(6.) ∫ x7
ln x dx here is the problem
Let u = ln x Let dv
= x7
du = (1/x)dx v = (1/8)x8
uv - ∫v du use the
parts formula
= (1/8)x8 ln x - (1/8)∫(x8)(1/x)dx
make substitutions
= (1/8)x8 ln x - (1/8)∫x7 dx
cancel
= (1/8)x8 ln x - (1/8)(1/8)x8 + C integrate
= (1/8)x8 ln x - (1/64)x8
+ C multiply
(7.) ∫ x sinh
x dx here is the problem
let u = x Let dv =
sinh x
du = dx v = cosh x
uv - ∫v du use the parts formula
= x cosh x - ∫cosh x dx make
substitutions
= x cosh x - sinh x + C integrate
(8.) ∫ x sin x
dx
here is the problem
let u = x Let dv = sin x
du = dx v = -cos x
uv - ∫v du use the parts formula
= -x cos x + ∫cos x dx make
substitutions
= -x cos x + sin x + C integrate
(9.) ∫ x[1 - (x/2)]1/2 dx
Let u =
x Let dv = [1 - (x/2)]1/2
du = dx v = -(4/3)[1 - (x/2)]3/2
uv - ∫v du use the parts formula
= (-4/3)x[1 - (x/2)]3/2 +
(4/3)∫[1 - (x/2)]3/2 dx
[make substitutions]
= (-4/3)x[1 - (x/2)]3/2 +
(4/3)(-2)(2/5)[1 - (x/2)]5/2 + C
[integrate]
= (-4/3)x[1 - (x/2)]3/2 -
(16/15)[1 - (x/2)]5/2 + C
multiply
(10.) ∫ x(3x +
1)1/2 dx
Let
u = x
du = dx
Let dv = (3x + 1)1/2
v = (2/9)(3x + 1)3/2
uv - ∫ v
du use the parts formula
= (2x/9)(3x + 1)3/2 - (2/9)∫(3x + 1)3/2 dx
make substitutions
= (2x/9)(3x + 1)3/2 -
(2/9)(1/3)(2/5)(3x + 1)5/2 + C
integrate
= (2x/9)(3x + 1)3/2 -
(4/135)(3x + 1)5/2 + C
multiply
(11.) ∫x2
cosh 2x dx
Let
u = x2
du = 2x dx
Let dv = cosh 2x
v = (1/2)sinh 2x
uv -∫v du use the parts formula
= (1/2)x2 sinh 2x - ∫x sinh 2x dx make substitutions
Let u = x
du = dx
Let dv = sinh 2x
v = (1/2)cosh 2x
= (1/2)x2 sinh 2x - [uv - ∫v du] use parts again
= (1/2)x2 sinh 2x - (1/2)x +
(1/2)∫ cosh 2x dx
[make substitutions]
= (1/2)x2 sinh 2x - (1/2)x + (1/2)(1/2)sinh 2x + C integrate
= (1/2)x2 sinh 2x - (1/2)x + (1/4)sinh 2x + C multiply
(12.) ∫ x2
cos 2x dx
(13.) ∫ cos (ln
x) dx
Let
u = ln x
x = eu
dx = eu du
∫ cos u *
eu du make substitutions
let w = cos u
dw = -sin u du
Let dv = eu
v = eu
wv - ∫v dw use the parts formula
= eu cos u + ∫eu sin u du
make substitutions
Let w = sin u
dw = cos u du
Let dv = eu
v = eu
= eu cos u + [wv - ∫v dw]
= eu cos u + eu
sin u - ∫eu cos u du
make substitutions
2∫eu cos u du = eu cos u + eu sin
u
∫eu
cos u du = (1/2)eu cos u + (1/2) eu sin u
∫elnx
cos (ln x)(1/x) replace u with ln x and
du with 1/x
∫cos (ln
x) cancel
= (1/2)x cos (ln x) + (1/2) x sin (ln
x) make substitutions
(14.) ∫
(ln x)2 dx
Let u = (ln x)2
du = (2/x)(ln x)dx
Let dv = dx
v = x
uv - ∫v du use the parts
formula
= x(ln x)2 - 2∫(ln x)dx make
substitutions
Let u = ln x
du = dx/x
Let dv = dx
v = x
= x(ln x)2 - 2[uv - ∫v du] use the parts
formula again
= x(ln x)2 - 2x ln x + 2∫dx make substitutions
= x(ln x)2 - 2x ln x + 2x +
C integrate
(15.) ∫
x5 ex^3 dx