NDA Integration: Standard Integrals, Substitution, Parts and Definite Integrals

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Integration as inverse differentiation; integration by substitution and by parts; standard integrals involving algebraic, trigonometric, exponential and related functions within syllabus scope; definite integrals and elementary area applications.

Integration as inverse differentiation; integration by substitution and by parts; standard integrals involving algebraic, trigonometric, exponential and related functions within syllabus scope; definite integrals and elementary area applications.

Differentiation, algebraic manipulation, trigonometric identities and functions.

• Antiderivative
• Indefinite integral
• Definite integral
• Constant of integration

• int x^n dx=x^(n+1)/(n+1)+C, n!=-1
• int 1/x dx=ln|x|+C
• int e^x dx=e^x+C
• int sin x dx=-cos x+C; int cos x dx=sin x+C
• Integration by parts: int u dv = uv - int v du
• Definite integral: int_a^b f(x)dx=F(b)-F(a) when F'=f

• Indefinite integrals require +C.
• Power rule excludes n=-1.
• Substitution should account for the differential factor.
• By-parts method is useful when choosing u and dv simplifies the remaining integral.
• For geometric area, split intervals where the integrand changes sign if the question asks for total area.

1. int (3x^2+2)dx=x^3+2x+C.
2. int x e^x dx: choose u=x, dv=e^x dx, giving x e^x - e^x + C.
3. int_0^1 2x dx=[x^2]_0^1=1.

• Dropping +C.
• Using power rule on 1/x.
• Choosing by-parts components that make the integral harder.
• Not changing limits or back-substituting consistently after definite-integral substitution.
• Calling a negative definite integral a negative geometric area.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Look first for a direct standard form or an inner derivative suggesting substitution. Use by parts for products where one factor simplifies when differentiated. In definite integrals, preserve exact values until the final step. If the item is really a differential-equation problem, switch to the dedicated method rather than forcing integration alone.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.