Integrals containing trigonometric functions or logarithms can be simplified by an appropriate substitution.
Example ① ∫ tan x dx
Solution:= ∫ (sin x / cos x) dx. Let u = cos x, du = −sin x dx.
= −∫ (1/u) du = −log|cos x| + C
Solution:= ∫ (sin x / cos x) dx. Let u = cos x, du = −sin x dx.
= −∫ (1/u) du = −log|cos x| + C
Example ② ∫ (log x)/x dx
Solution:Let u = log x, du = dx/x.
∫ u du = u²/2 + C = (log x)²/2 + C
Solution:Let u = log x, du = dx/x.
∫ u du = u²/2 + C = (log x)²/2 + C
Key Point
- Form f'/f → log|f|
- For forms containing log or an exponential, use substitution to set the function equal to u
Practice Problems
- ∫ (2x+3)/(x²+3x+5) dx
Answer and Explanation
- Answer:log|x²+3x+5| + C
The numerator has the form of the derivative of the denominator.