Integrate ≤ft(a x + \; {b}{x³} + \; {c}{x^{7}}) x²

Correct Answer is : 14ax4+blog⁡x− 14cx−4+k\; 1/4 a x^{4} + b x - \; 1/4 c x^{-4} + k41ax4+blogx−41cx−4+k

\; 1/4 a x^{4} + b x - \; 1/4 c x^{-4} + k

4 a x^{4} + b x - 4 c x^{-4} + k

\; 1/4 a x^{4} + b x + \; 1/4 c x^{-4} + k

Correct Answer is : 14ax4+blog⁡x− 14cx−4+k\; 1/4 a x^{4} + b x - \; 1/4 c x^{-4} + k41ax4+blogx−41cx−4+k

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Integral Calculus

Question : 53 of 73

Marks: +1, -0

Integrate

\left(a x + \; \frac{b}{x^{3}} + \; \frac{c}{x^{7}}\right) x^{2}

Solution:

Concept:

Simplify the integrand by multiplying through by

x^2

, then integrate term by term using standard power rule and log rule.

Explanation:

First, simplify the integrand:

\left(ax + \frac{b}{x^3} + \frac{c}{x^7}\right) x^2 = a x \cdot x^2 + \frac{b}{x^3} \cdot x^2 + \frac{c}{x^7} \cdot x^2 = a x^3 + b x^{-1} + c x^{-5}

Now integrate each term:

\int a x^3 \, dx = a \cdot \frac{x^{4}}{4} = \frac{a}{4} x^4

\int b x^{-1} \, dx = b \ln|x|

(we write

\log x

as common notation).

\int c x^{-5} \, dx = c \cdot \frac{x^{-4}}{-4} = -\frac{c}{4} x^{-4}

Thus the integral is

\frac{1}{4} a x^4 + b \log x - \frac{1}{4} c x^{-4} + k

Answer:

\frac{1}{4} a x^4 + b \log x - \frac{1}{4} c x^{-4} + k

(Option A).

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