AHL 5.11—Indefinite integration, reverse chain, by substitution Questionbank

A function $k(x)$ has derivative $k^{\prime}(x)=\frac{3 \cos x}{\sin ^{2} x+1}$. The graph of $k$ passes through the point $\left(\frac{\pi}{2}, 0\right)$.

A function $h(x)$ has derivative $h^{\prime}(x)=\frac{8 x}{\left(x^{2}+4\right)^{2}}$. The graph of $h$ passes through $(0,1)$.

Let $f^{\prime}(x)=\frac{\cos x}{\sin ^{2} x+4}$. The region $R$ is enclosed by the graph of $f^{\prime}$, the x -axis, and the lines $x=0$ and $x=\pi$.

A function $h(x)$ has second derivative $h^{\prime \prime}(x)=\frac{8 x}{\left(x^{2}+1\right)^{3}}$, for $x \in \mathbb{R}$. Given $h^{\prime}(1)=2$ and $h(1)=3$.

The derivative of a function $f$ is given by $f^{\prime}(x)=\frac{4 e^{2 x}}{e^{2 x}+1}$. The graph of $f$ passes through the point $(0,-1)$.

A function $g(x)$ has derivative $g^{\prime}(x)=\frac{5 \sin x}{\cos ^{2} x+2}$, for $x \in \mathbb{R}$. The graph of $g$ passes through $(0,1)$.

The region bounded by the curve $y=\frac{3 x}{\sqrt{x^{2}+1}}$, the x-axis, and the lines $x=0$ and $x=3$ is rotated through $2 \pi$ radians about the x-axis to form a solid of revolution.

Consider the function $y=\sqrt{x^{2}+2 x}$. Let $f(x)=\sqrt{x^{2}+2 x}$ and $g(x)=\frac{4-x}{\sqrt{x^{2}+2 x}}$, for $x \geq 0$. The region $R$ is enclosed by the graphs of $f, g$, the y -axis, and the line $x=2$.

Given that $\int_{-1}^{3} f(x) d x=8$ and $\int_{0}^{3} f(x) d x=5$, find

The rate of change of a population $P(t)$ of a species, in thousands, is given by $\frac{d P}{d t}=\frac{4 t}{\sqrt{t^{2}+9}}$, where $t \geq 0$ is time in years. At $t=0$, the population is 10 thousand.