Contents

8: Techniques of Integration

8.1 Basic Integration Formulas

TABLE 8.1 Basic integration formulas
$$\int{du}=u+C\tag{1}$$
$$\int{k \;du}=k u+C \quad\text{(any number k)} \tag{2}$$
$$\int{du + dv}=\int{du} +\int{dv} \tag{3}$$
$$\int{u^n du}=\frac{u^{n+1}}{n+1} +C \quad (n\neq -1)\tag{4}$$
$$\int{\frac{du}{u}}=ln |u| +C \tag{5}$$

8.4 Trigonometric Integrals

Products of Powers of Sines and Cosines

We begin with integrals of the form: $$\int sin ^m x cos ^n x dx$$ where m and n are nonnegative integers (positive or zero). We can divide the work into three cases.
Case 1 If m is odd, we write m as 2k + 1 and use the identity sin²x = 1 - cos²x to obtain $$sin ^m x= sin ^{2k+1} x = (sin ^2 x)^k sin x = (1-cos^2 x)^ksinx \tag{1}$$ Then we combine the single sin x with dx in the integral and set sin x dx equal to -d(cos x).
Case 3 If both m and n are even in ∫ sin ^m x cos ⁿ x dx, we sustitute $$sin ^2 x = \frac{1-cos 2x}{2} , \; cos ^2 x = \frac{1+cos 2x}{2}\tag{2}$$ to reduce teh integrand to one in lower powers of cos 2x.