Integration by recognition and identities

What you’ll learn

• Integrate standard exponential and trigonometric functions.
• Recognise reverse-chain-rule forms.
• Use identities to simplify an integrand.

Understand the idea

A constant multiple inside a function becomes a divisor in its standard integral. For example, integrating $\cos(kx)$ gives $\sin(kx)/k$. This follows by differentiating the proposed antiderivative. Similarly, $\int a^x\,dx=a^x/\ln a+C$ for $a>0$, $a\ne1$. For logarithmic integrals, use $\ln|x|$ where the original expression may be defined for negative inputs.

Recognition means spotting an inner function together with its derivative. An integrand proportional to $f'(x)/f(x)$ has a logarithmic antiderivative. An integrand $f'(x)[f(x)]^n$ has a power antiderivative for $n\ne-1$. Check the multiplier by differentiation.

Some trigonometric integrands need an identity first. Replace $\sin^2x$ by $(1-\cos2x)/2$, $\cos^2x$ by $(1+\cos2x)/2$, or $\tan^2x$ by $\sec^2x-1$. These forms reduce the problem to standard integrals.

Key formulae

$$\int e^{kx}\,dx=\frac1k e^{kx}+C\quad(k\ne0)$$

$$\int\cos(kx)\,dx=\frac1k\sin(kx)+C,\qquad \int\sin(kx)\,dx=-\frac1k\cos(kx)+C$$

$$\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C$$

$$\int f'(x)[f(x)]^n\,dx=\frac{[f(x)]^{n+1}}{n+1}+C\quad(n\ne-1)$$

Worked example

Find $\int\dfrac{6x}{3x^2+4}\,dx$.

1. The denominator is $f(x)=3x^2+4$.
2. Its derivative is $f'(x)=6x$, exactly the numerator.
3. Apply the logarithmic recognition rule. The denominator is positive for every real $x$.

Answer: $\ln(3x^2+4)+C$.