IB Maths AA HL Sl 5 10 Indefinite Integration Questions

Practise integrating powers, trigonometric, reciprocal and exponential functions, including composite expressions with substitution or the reverse chain rule.

Exam points

Find antiderivatives of power, logarithmic, exponential and trigonometric functions, including a constant of integration.
Use substitution and integration by parts to integrate composite, product and rational expressions, including definite and improper integrals.
Use an initial condition or a known point on a curve to determine the integration constant and hence find the function or curve equation.
Solve separable or otherwise given differential equations by integration and apply the resulting constants or conditions.
Use integration and series-related antiderivatives to obtain exact values or approximations where the required expansion is given.

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Study SL 5.10—Indefinite integration conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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The functions f and g are defined by

\begin{aligned} & f(x)=\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \\ & g(x)=\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \end{aligned}

Use the substitution $u=\mathrm{e}^{x}$ to find $\int_{0}^{\ln 3} \frac{1}{4 f(x)-2 g(x)} \mathrm{d} x$ . Give your answer in the form $\frac{\pi \sqrt{a}}{b}$ where $a, b \in \mathbb{Z}^{+}$ .

$\quad u=\mathrm{e}^{x} \Rightarrow \mathrm{~d} u=\mathrm{e}^{x} \mathrm{~d} x$

\int \frac{\mathrm{e}^{x}}{\mathrm{e}^{2 x}+3} \mathrm{~d} x=\int \frac{1}{u^{2}+3} \mathrm{~d} u

(when x=0, u=1 and when $x=\ln 3, u=3$ )

\begin{aligned} & \int_{1}^{3} \frac{1}{u^{2}+3} \mathrm{~d} u=\left[\frac{1}{\sqrt{3}} \arctan \left(\frac{u}{\sqrt{3}}\right)\right]_{1}^{3} \\ & \left.=\left[\frac{1}{\sqrt{3}} \arctan \left(\frac{\mathrm{e}^{x}}{\sqrt{3}}\right)\right]_{0}^{\ln 3}\right) \\ & =\frac{\pi \sqrt{3}}{9}-\frac{\pi \sqrt{3}}{18} \\ & =\frac{\pi \sqrt{3}}{18} \end{aligned}

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IB Maths AA HL Sl 5 10 Indefinite Integration Questions question 1