(a) If \(y = (2x + 3)^{7} + \frac{x + 1}{2x - 1}\), find the value of \(\frac{\mathrm d y}{\mathrm d x}\) at x = -1. (b) Using the substitution, \(u = x + 2\), evaluate \(\int_{1} ^{2} \frac{x - 1}{(x + 2)^{4}} \mathrm d x\).

Take Free Practice Test On 2025 JAMB UTME, Post UTME, WAEC SSCE, GCE, NECO SSCE

(a) If

y = (2 x + 3)^{7} + \frac{x + 1}{2 x - 1}

, find the value of

\frac{d y}{d x}

at x = -1.
(b) Using the substitution,

u = x + 2

, evaluate

\int_{1}^{2} \frac{x - 1}{(x + 2)^{4}} d x

Explanation

(a)

y = (2 x + 3)^{7} + \frac{x - 1}{2 x - 1}

\frac{d y}{d x} = 7 (2 x + 3)^{6} . 2 + \frac{(2 x - 1) .1 - (x - 1) .2}{(2 x - 1)^{2}}

14 (2 x + 3)^{6} + \frac{2 x - 1 - 2 x + 2}{(2 x - 1)^{2}}

14 (2 x + 3)^{6} + \frac{1}{(2 x - 1)^{2}}

At x = -1,

\frac{d y}{d x} = 14 (- 1)^{6} + \frac{1}{(2 (- 1) - 1)^{2}}

\frac{d y}{d x} = 14 + \frac{1}{9} = 14 \frac{1}{9}

(b)

\int_{1}^{2} \frac{x - 1}{(x + 2)^{4}} d x

u = x + 2 ⟹ d u = d x

If x = 1, u = 1 + 2 = 3; x = 2, u = 2 + 2 = 4.

\int_{1}^{2} \frac{x - 1}{(x + 2)^{4}} d x = \int_{3}^{4} \frac{u - 3}{u^{4}}

\int_{3}^{4} (\frac{u}{u^{4}} - \frac{3}{u^{4}}) d u

\int_{3}^{4} (u^{- 3} - 3 u^{- 4}) d u

[\frac{u^{- 2}}{- 2} + u^{- 3}]_{3}^{4}

[\frac{1}{u^{3}} - \frac{1}{2 u^{2}}]_{3}^{4}

(\frac{1}{4^{3}} - \frac{1}{2 (4^{2})}) - (\frac{1}{3^{3}} - \frac{1}{2 (3^{2})})

\frac{1}{64} - \frac{1}{32} - \frac{1}{27} + \frac{1}{18}

\frac{5}{1728}

Prev Current Next

Register • Login

(a) If $y = (2 x + 3)^{7} + \frac{x + 1}{2 x - 1}$ , find the value of \(\frac{\mathrm d ...

Trending Questions

Explanation

Subjects

• Accounts - Principles of Accounts

• Animal Husbandry

• Agricultural Science

• Arabic

• Biology

• Book Keeping

• Chemistry

• Commerce

• Christian Religious Knowledge (CRK)

• Civic Education

• Catering Craft Practice

• Computer Studies

• Data Processing

• English language

• Economics

• Fine Arts

• French

• Further Mathematics

• Geography

• Government

• History

• Hausa

• Home Economics

• Insurance

• Igbo

• Islamic Religious Knowledge (IRK)

• Literature in English

• Music

• Marketing

• Mathematics

• Physics

• Yoruba

Register • Login

(a) If y=(2x+3)7+x+12x−1, find the value of \(\frac{\mathrm d ...

Trending Questions

Explanation

Subjects

• Accounts - Principles of Accounts

• Animal Husbandry

• Agricultural Science

• Arabic

• Biology

• Book Keeping

• Chemistry

• Commerce

• Christian Religious Knowledge (CRK)

• Civic Education

• Catering Craft Practice

• Computer Studies

• Data Processing

• English language

• Economics

• Fine Arts

• French

• Further Mathematics

• Geography

• Government

• History

• Hausa

• Home Economics

• Insurance

• Igbo

• Islamic Religious Knowledge (IRK)

• Literature in English

• Music

• Marketing

• Mathematics

• Physics

• Yoruba

(a) If $y = (2 x + 3)^{7} + \frac{x + 1}{2 x - 1}$ , find the value of \(\frac{\mathrm d ...