Formula for Integration by Parts
If u and v are two functions of x, then the formula for integration by parts is –
∫ u.v dx = u ∫ v dx – ∫[dudx.∫v dx]dx
i.e The integral of the product of two functions = (first function) × (Integral of Second function) – Integral of { (Diff. of first function) × (Integral of Second function)}
Note – We can choose the first function as the function which comes first in the word ILATE, where
I – Stands for the Inverse Trigonometric Function
L – Stands for the Logarithmic Function
A – Stands for the Algebraic Function
T – Stands for the Trigonometric Function
E – Stands for the Exponential Function
Example : Solve the integral ∫ xsin3x dx using the formula for integration by parts.
Solution : Here both the functions viz. x and sin3x are easily integrable and the derivative of x is one, a less complicated function. Therefore, we take x as the first function and sin3x as the second function.
∴ I = ∫ x cos3x dx
= x [∫ sin3x dx] – ∫[ddx(x) × ∫sin3x dx] dx
= x (−13 cos3x) – ∫ [−13 cos3x] dx
⟹ I = −13 xcos3x + 13 ∫ cos3x dx
⟹ I = -13 xcos3x + 19 sin3x + C
Example : Solve the integral ∫ xsin−1x dx using the formula for integration by parts.
Solution : Taking sin−1x as the first function and x as the second function by using the ILATE rule.
I = ∫ xsin−1x
= (sin−1x)x22 – ∫[1√1−x2 × x22] dx
= x22(sin−1x) + 12 ∫[(−x)2√1−x2 dx = x22(sin−1x) + 12 ∫[1−x2−1√1−x2 dx
⟹ I = x22(sin−1x) + 12 {∫[1−x2√1−x2 dx – 1√1−x2 dx}
⟹ I = x22(sin−1x) + 12 {∫[√1−x2 dx – 1√1−x2 dx}
I = x22(sin−1x) + 12 {12x[√1−x2 dx + 12 sin−1x – sin−1x] + C
⟹ I = x22(sin−1x) + 14x√1−x2 dx – 14 sin−1x + C
Hope you learnt what is the formula for integration by parts, learn more concepts of Indefinite Integration and practice more questions to get ahead in the competition. Good luck!