5.11 Further integrationIB Maths: Applications and Interpretation HL: Revision notes

Integration reverses differentiation. Include $+c$ for an indefinite integral.

• $\int x^n\,dx=\frac{x^{n+1}}{n+1}+c$ for $n\in\mathbb{Q}$, $n\neq-1$. For example $\int6\sqrt t\,dt=4t^{3/2}+c$.
• $\int\frac1x\,dx=\ln|x|+c$ (the case $n=-1$)
• $\int\sin x\,dx=-\cos x+c$ and $\int\cos x\,dx=\sin x+c$
• $\int\frac{1}{\cos^2x}\,dx=\tan x+c$
• $\int e^x\,dx=e^x+c$ Rewrite roots and fractions as powers first: $\frac{5}{x^2}=5x^{-2}$ integrates to $-5x^{-1}+c$.

If $F'(x)=f(x)$, then for constants $a\neq0$ and $b$: $\int f(ax+b)\,dx=\frac{1}{a}F(ax+b)+c.$ Divide by the coefficient of $x$.

• $\int\sin(2x+5)\,dx=-\frac12\cos(2x+5)+c$
• $\int\frac{1}{3x+2}\,dx=\frac13\ln|3x+2|+c$
• $\int e^{-0.5t}\,dt=-2e^{-0.5t}+c$ This is integration by inspection: you spot the chain rule that has been used and undo it.

For an integral of the form $\int f(g(x))\,g'(x)\,dx$, let $u=g(x)$, so $du=g'(x)\,dx$. Then it becomes $\int f(u)\,du$. Example: $\int4x\sin(x^2)\,dx$. Let $u=x^2$, $du=2x\,dx$. Then $\int2\sin u\,du=-2\cos u+c=-2\cos(x^2)+c$. Example: $\int\frac{\sin x}{\cos x}\,dx$. Let $u=\cos x$, $du=-\sin x\,dx$. Then $\int\frac{-1}{u}\,du=-\ln|\cos x|+c$. Always return to the original variable in an indefinite integral. The method works when the derivative of the inner function appears, up to a constant multiple.

$\int_a^bf(x)\,dx=F(b)-F(a).$ No constant is needed. With substitution you can change the limits: for $\int_0^1 2xe^{x^2}\,dx$ with $u=x^2$, the limits become $u=0$ to $u=1$, so $\int_0^1e^u\,du=e-1$. Alternatively, find the antiderivative in $x$ and use the original limits. Use your GDC to check numerical answers. If asked for an exact value, give logarithms and trigonometric values exactly, using $\ln\frac{1}{\sqrt2}=-\frac12\ln2$.

A rate of change integrates to a total. If $\frac{dV}{dt}=r(t)$, then the change in $V$ between $t_1$ and $t_2$ is $\int_{t_1}^{t_2}r(t)\,dt$. Use a given value, such as $P(0)=5$, to find the constant. Example: $\frac{dP}{dt}=\frac{12}{3t+2}$ gives $P=4\ln(3t+2)+c$. With $P(0)=5$, $c=5-4\ln2$. Always give units and state what the answer means in context.