(1.) ∫xe^3x dx here is the problem

Let u = x Let dv = e^3x

du = dx v = (1/3)e^3x

uv - ∫v du use the parts formula

= (1/3)xe^3x - (1/3)∫e^3x dx make substitutions

= (1/3)xe^3x - (1/3)(1/3)e^3x + C integrate

= (1/3)xe^3x - (1/9)e^3x + 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.) ∫x²e^x/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 = e^u

dx = e^u du

∫ cos u * e^u du make substitutions


let w = cos u

dw = -sin u du

Let dv = e^u

v = e^u

wv - ∫v dw use the parts formula

= e^u cos u + ∫e^u sin u du make substitutions

Let w = sin u

dw = cos u du

Let dv = e^u

v = e^u

= e^u cos u + [wv - ∫v dw]

= e^u cos u + e^u sin u - ∫e^u cos u du make substitutions
2∫e^u cos u du = e^u cos u + e^u sin u

∫e^u cos u du = (1/2)e^u cos u + (1/2) e^u sin u

∫e^lnx 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 e^{x^3} dx