Find {x³+5x+1}{x³} dx

Correct Answer is : x−5x−12x2+cx - 5/x - {1}{2x²} + cx−x5−2x21+c

Correct Answer is : x−5x−12x2+cx - 5/x - {1}{2x²} + cx−x5−2x21+c

Test Index

WAEC Further Mathematics 2015 Paper

Question : 27

Total: 40

Find

\int \frac{x^{3}+5x+1}{x^{3}} dx

Solution:

Concept:

Simplify the integrand by dividing each term by

x^3

, then integrate using the power rule.

Explanation:

Rewrite the fraction as separate terms:

\frac{x^3+5x+1}{x^3} = 1 + \frac{5}{x^2} + \frac{1}{x^3}

Express the terms in index form:

\int \left(1 + 5x^{-2} + x^{-3}\right) dx

Integrate term by term:

\int 1\,dx = x

\int 5x^{-2}\,dx = 5 \cdot \frac{x^{-1}}{-1} = -5x^{-1}

\int x^{-3}\,dx = \frac{x^{-2}}{-2} = -\frac{1}{2}x^{-2}

Add the constant of integration

c

So the integral becomes:

x - 5x^{-1} - \frac{1}{2}x^{-2} + c

This simplifies to:

x - \frac{5}{x} - \frac{1}{2x^2} + c

Answer:

Option D:

x - \frac{5}{x} - \frac{1}{2x^2} + c

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