integrate the function (1-x)(2+3x)(5-4x)
Answers
EXPLANATION.
⇒ ∫(1 - x)(2 + 3x)(5 - 4x)dx.
As we know that,
Factorizes the equation, we get.
⇒ (1 - x)(2 + 3x))(5 - 4x).
⇒ (1 - x)[10 - 8x + 15x - 12x²].
⇒ (1 - x)[10 + 7x - 12x²].
⇒ 10 + 7x - 12x² - 10x - 7x² + 12x³.
⇒ 10 - 3x - 19x² + 12x³.
⇒ ∫(10 - 3x - 19x² + 12x³)dx.
⇒ ∫(10)dx - ∫(3x)dx - ∫(19x²)dx + ∫(12x³)dx.
Using the formula,
⇒ ∫aⁿdx = aⁿ⁺¹/n + 1.
Take constant term outside the integration, we get.
⇒ 10∫dx - 3∫xdx - 19∫x²dx + 12∫x³dx.
⇒ 10x - 3(x²/2) - 19(x³/3) + 12(x⁴/4) + c.
⇒ 10x - 3x²/2 - 19x³/3 + 3x⁴ + c.
MORE INFORMATION.
Standard integrals.
(1) = ∫Sin(x)dx = - Cos(x) + c.
(2) = ∫Cos(x)dx = Sin(x) + c.
(3) = ∫tan(x)dx = ㏒(Sec(x)) + c = - ㏒(Cos(x)) + c.
(4) = ∫cot(x)dx = ㏒(sin(x)) + c.
(5) = ∫Sec(x)dx = ㏒(Sec(x) + tan(x)) + c. = -㏒(Sec(x) - tan(x)) + c = ㏒ tan(π/4 + x/2) + c.
(6) = ∫Cosec(x)dx = -㏒(Cosec(x) + Cot(x)) + c = ㏒(Cosec(x) - Cot(x)) + c = ㏒ tan(x/2) + c.
(7) = ∫Sec(x).tan(x)dx = Sec(x) + c.
(8) = ∫Cosec(x).Cot(x)dx = -Cosec(x) + c.
(9) = ∫Sec²xdx = tan(x) + c.
(10) = ∫Cosec²xdx = -Cot(x) + c.
EXPLANATION.
⇒ ∫(1 - x)(2 + 3x)(5 - 4x)dx.
As we know that,
Factorizes the equation, we get.
⇒ (1 - x)(2 + 3x))(5 - 4x).
⇒ (1 - x)[10 - 8x + 15x - 12x²].
⇒ (1 - x)[10 + 7x - 12x²].
⇒ 10 + 7x - 12x² - 10x - 7x² + 12x³.
⇒ 10 - 3x - 19x² + 12x³.
⇒ ∫(10 - 3x - 19x² + 12x³)dx.
⇒ ∫(10)dx - ∫(3x)dx - ∫(19x²)dx + ∫(12x³)dx.
Using the formula,
⇒ ∫aⁿdx = aⁿ⁺¹/n + 1.
Take constant term outside the integration, we get.
⇒ 10∫dx - 3∫xdx - 19∫x²dx + 12∫x³dx.
⇒ 10x - 3(x²/2) - 19(x³/3) + 12(x⁴/4) + c.
⇒ 10x - 3x²/2 - 19x³/3 + 3x⁴ + c.
MORE INFORMATION.
Standard integrals.
(1) = ∫Sin(x)dx = - Cos(x) + c.
(2) = ∫Cos(x)dx = Sin(x) + c.
(3) = ∫tan(x)dx = ㏒(Sec(x)) + c = - ㏒(Cos(x)) + c.
(4) = ∫cot(x)dx = ㏒(sin(x)) + c.
(5) = ∫Sec(x)dx = ㏒(Sec(x) + tan(x)) + c. = -㏒(Sec(x) - tan(x)) + c = ㏒ tan(π/4 + x/2) + c.
(6) = ∫Cosec(x)dx = -㏒(Cosec(x) + Cot(x)) + c = ㏒(Cosec(x) - Cot(x)) + c = ㏒ tan(x/2) + c.
(7) = ∫Sec(x).tan(x)dx = Sec(x) + c.
(8) = ∫Cosec(x).Cot(x)dx = -Cosec(x) + c.
(9) = ∫Sec²xdx = tan(x) + c.
(10) = ∫Cosec²xdx = -Cot(x) + c.