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What is the Formula for Integration by Parts ?

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 xsin1x dx using the formula for integration by parts.

Solution : Taking sin1x as the first function and x as the second function by using the ILATE rule.

I = xsin1x

= (sin1x)x22[11x2 × x22] dx

= x22(sin1x) + 12 [(x)21x2 dx = x22(sin1x) + 12 [1x211x2 dx

I = x22(sin1x) + 12 {[1x21x2 dx – 11x2 dx}

I = x22(sin1x) + 12 {[1x2 dx – 11x2 dx}

I = x22(sin1x) + 12 {12x[1x2 dx + 12 sin1xsin1x] + C

I = x22(sin1x) + 14x1x2 dx – 14 sin1x + 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!

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