Integration by Parts + Partial Fractions
Indefinite Integration
50
JEE Qs
15%
Hard
90
min
Master the systematic application of ILATE and partial fraction decomposition types; meticulous algebraic manipulation is key to avoiding errors and arriving at correct solutions.
🧮 Key Formulas
✅ Key Points for JEE
- 1For Integration by Parts, use the 'ILATE' rule (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential) to prioritize 'u' (function to be differentiated) and the remaining as 'dv' (function to be integrated).
- 2Recognize cyclic integrals (e.g., ∫ e^(ax) sin(bx) dx) where Integration by Parts is applied twice to obtain an equation for the original integral.
- 3For Partial Fractions, always check if the rational function is proper (degree of numerator < degree of denominator). If improper, first perform polynomial long division to get a polynomial + a proper rational function.
- 4Correctly decompose the denominator into its factors: distinct linear, repeated linear, or irreducible quadratic factors, as the form of the partial fraction decomposition depends entirely on this.
- 5Efficiently solve for the constants (A, B, C, etc.) in partial fractions by either substituting suitable values of x (especially roots of linear factors) or by comparing coefficients of powers of x.
⚠️ Common Mistakes
- ✕Incorrect application of the 'ILATE' rule, leading to a more complex integral in Integration by Parts.
- ✕Algebraic and sign errors in the `uv - ∫ v du` step, especially with the negative sign inside the integral.
- ✕Failing to divide improper rational functions before applying partial fraction decomposition.
- ✕Incorrectly setting up the partial fraction decomposition, particularly for repeated linear factors or irreducible quadratic factors.
- ✕Calculation errors when solving the system of equations for the constants A, B, C, etc., in partial fractions.
📝 Practice Questions
See allQ4. Let ∫x3 sin x dx = g(x) + C , where C is the constant of integration. If 8 (g ( π2 ) + g′ ( π2 )) = απ3 + βπ2 + γ, α, β, γ ∈Z , then α + β −γ equals : (1) 48 (2) 55 (3) 62 (4) 47
Q15.If f(x) = ∫ 1 dx, f(0) = −6, then f(1) is equal to : x1/4(1+x1/4) (1) 4 (loge 2 −2) (2) 2 −loge2 2 (3) loge 2 + 2 (4) 4 (loge 2 + 2)
Q6. x sin−1 x sin−1 x x 1 + If ∫ex + 1−x2 = g(x) + C, where C is the constant of integration, then g ( 2 ) equals (1−x2)3/2 ( √1−x2 )dx : (1) π (2) π 4 √e3 6 √e3 (3) π 4 √e2 (4) π6 √e2
Q19.Let I(x) = ∫ 11 15 . If I(37) −I(24) = 4 1 − 1 b, c ∈N (x−11) 13 (x+15) 13 ( b 13 c 13 ), (1) 22 (2) 39 (3) 40 (4) 26
Q22.If ∫2x2+5x+9 dx = x√x2 + x + 1 + α√x2 + x + 1 + β loge x + 12 + √x2 + √x2+x+1 constant of integration, then α + 2β is equal to _______.
Q88.If ∫ B 1 dx = A( αx−1βx+3 ) 5√(x−1)4(x+3)6 α + β + 20AB is__________
NCERT Chapters
- Class 12 Mathematics Ch 7: Integrals