IB Mathematics Applications & Interpretation Topic…

A function $f(x)$ has derivative $f^{\prime}(x)=\frac{6 x^{2}}{x^{3}+8}$ , for . The graph of passes through the point .

(a) Find $\int \frac{6 x^{2}}{x^{3}+8}\,dx$ .

(b) Hence, find $f(x)$ .

(e) the total distance travelled, including both forward and backward motion, in the interval $0\leq t\leq 3$ .

Given that $\int_{-2}^2 g(y) \, \mathrm{d}y = 10$ and $\int_0^2 g(y) \, \mathrm{d}y = 12$ , find

(a) $\int_{-2}^0 (g(y) + 2) \, \mathrm{d}y$ .

(b) $\int_{-2}^0 g(y+2) \, \mathrm{d}y$ .

Let $y=\sin^5(2x)$ .

Consider the function $g(x)=3-2\sin^4(2x)\cos(2x)$ , defined for $0\le x\le \frac{\pi}{4}$ .

The region $S$ is bounded by the graph of $g$ , the $x$ -axis, the $y$ -axis, and the line $x=\frac{\pi}{4}$ .

(b) Using your result from the previous part, find $\int \sin^4(2x)\cos(2x)\,dx$ .

(a) Find $\int t^2\mathrm{e}^{8-t^3}\,\mathrm{d}t$ .

(b) Given that $g(2)=4$ , find an expression for $g(t)$ .

The velocity of a particle, $v$ m s $^{-1}$ , is modelled by the function $v(t) = -3t^2 + 18t - 24$ for $t \geq 0$ , where $t$ is the time in seconds.

(a) Find the values of $t$ for which the velocity is zero.

(b) Find $|v'(4)|$ .

(d) Assuming the particle is at the origin at $t=0$ , find an expression for the displacement $s$ (in metres) of the particle at time $t$ seconds.

(a) the derivative $\frac{dy}{dx}$ .

(d) the precise area of $S$ .

IB Mathematics Applications & Interpretation AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Applications & Interpretation AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Applications & Interpretation AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Applications & Interpretation AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank