Find $\displaystyle \displaystyle\int \cos (3 \ln x) \, \mathrm{d}x$.[4] Let $I$ be the ind

Question

Find $\displaystyle \displaystyle\int \cos (3 \ln x) \, \mathrm{d}x$.[4] Let $I$ be the indefinite integral $\displaystyle \displaystyle\int \frac{\mathrm{P}(x)}{1-\sqrt{x}} \, \mathrm{d}x$, $0 < x < 1$, where $\mathrm{P}(x)$ is a polynomial in $x$. Find $I$ when $\mathrm{P}(x) = 1 - x$.[2] By using the substitution $u = 1 - \sqrt{x}$, find $I$ when $\mathrm{P}(x) = 1$.[3] Hence find $I$ when $\mathrm{P}(x) = x$.[2]

Accepted Answer

(a) $\frac{x}{10}\cos(3\ln x)+\frac{3x}{10}\sin(3\ln x)+c$. (b)(i) $x+\frac23x^{3/2}+c$; (ii) $-2\sqrt{x}-2\ln(1-\sqrt{x})+c$. Hence, for $\mathrm P(x)=x$, $-x-\frac23x^{3/2}-2\sqrt{x}-2\ln(1-\sqrt{x})+c$.

2025 NJC P1 Q9

2025 NJC P1 Q9

Video Solution:

Solution:

Solution locked