Reverse exponential differentiation and apply initial values. Integrate trigonometric functions in radians with the correct sign. Recognise logarithmic primitives and their domain.
Notes and key methods
For \(\displaystyle k\ne 0, \int Ae^{kx}dx= (\frac{A}{k})e^{kx}+ C\). The coefficient is divided by k rather than multiplied. Use exact identities such as \(\displaystyle e^{\ln 3}=3\) before calculating.
An initial value fixes C; a definite integral uses the same primitive at both endpoints.
For \(\displaystyle k\ne 0, \int \cos (kx)dx= \frac{\sin (kx)}{k}+ C\) and \(\displaystyle \int \sin (kx)dx= - \frac{\cos (kx)}{k}+ C\). The negative sign in the sine primitive is checked by differentiating.
Use radian endpoints. When applying an initial condition, a cosine term at zero generally contributes a nonzero constant.
For \(\displaystyle b\ne 0, \int \frac{A}{bx+ c}dx= (\frac{A}{b})\ln |bx+ c|+ C\) on an interval that excludes the pole. Absolute values allow negative arguments of the inner linear function while keeping the real logarithm valid.
On an interval where bx+c>0, write ln(bx+c). A difference of endpoint logarithms can be combined as a logarithm of a positive ratio.
Check the conditions
Divide by the exponential inner multiplier.
Apply both endpoint values.
Divide by k and use radians.
Check the sign of the cosine primitive.
Exclude the pole from the interval.
Use a logarithm rather than the ordinary power rule at exponent −1.
Three worked examples
Example 1 · Exponential integration
The integrand is \(f(x)=2 e^{2 x}\), on real x.
Find the general indefinite integral of the given function.
Evaluate the definite integral from \(0\) to \(\displaystyle \frac{\log{\left(3 \right)}}{2}\).
Find the primitive G satisfying \(\displaystyle G(0)=0\).
Find the integral from 0 to \(\displaystyle \frac{\ln 4}{2}\).
Show the reasoning
(a)
Differentiate \(e^{2 x}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(e^{2 x} + C\)
(b)
Use upper primitive value minus lower primitive value: \(3\) − (\(1\)) = \(2\).
Answer: \(2\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(G(x)=e^{2 x} - 1\)
(d)
Subtract the exponential endpoint values.
\(3\)
Answer: \(3\).
Example 2 · Sine and cosine integrals
The integrand is \(f(x)=6 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\), on real x.
Find the general indefinite integral of the given function.
Evaluate the definite integral from \(0\) to \(\displaystyle \frac{\pi}{6}\).
Find the primitive G satisfying \(\displaystyle G(0)=0\).
Find the integral from 0 to \(\displaystyle \frac{\pi }{6}\).
Show the reasoning
(a)
Differentiate \(\sin{\left(3 x \right)} - 2 \cos{\left(3 x \right)}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(\sin{\left(3 x \right)} - 2 \cos{\left(3 x \right)} + C\)
(b)
Use upper primitive value minus lower primitive value: \(1\) − (\(-2\)) = \(3\).
Answer: \(3\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(G(x)=\sin{\left(3 x \right)} - 2 \cos{\left(3 x \right)} + 2\)
(d)
Evaluate sine and cosine at 0 and \(\displaystyle \frac{\pi }{2}\).
\(3\)
Answer: \(3\).
Example 3 · Reciprocal-linear integrals
The integrand is \(\displaystyle f(x)=\frac{4}{x + 3}\), on \(x>-3\).
Find the general indefinite integral of the given function.
Evaluate the definite integral from \(0\) to \(1\).
Find the primitive G satisfying \(\displaystyle G(0)=0\).
Find the x-coordinate of the pole.
Show the reasoning
(a)
Differentiate \(4 \log{\left(x + 3 \right)}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(4 \log{\left(x + 3 \right)} + C\)
(b)
Use upper primitive value minus lower primitive value: \(4 \log{\left(4 \right)}\) − (\(4 \log{\left(3 \right)}\)) = \(\displaystyle \log{\left(\frac{256}{81} \right)}\).