Is Integration part of the EAPCET Mathematics syllabus?

Yes. Integration is taught in Intermediate second year · Paper IIB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.

How many Mathematics questions are there in EAPCET?

80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.

What mistakes do students make in Integration?

Forgetting to divide by the coefficient of x in a linear inside function; ∫ cos 3x dx = (sin 3x)/3 + C, not sin 3x + C. Leaving out the 1/a in ∫ dx/(a² + x²); the result is (1/a) tan⁻¹(x/a) + C. In the eˣ [f + f′] pattern, writing eˣ [f + f′] as the answer; the answer is only eˣ f(x) + C. Choosing u badly in integration by parts, such as u = eˣ in ∫ x eˣ dx, which makes the integral harder; follow ILATE and take u = x.

EAPCET Mathematics · Intermediate second year · Paper IIB

Integration

Integration covers indefinite integrals: standard forms, substitution, integration by parts and partial fractions.

Subject: Mathematics
Taught in: Intermediate second year · Paper IIB
Mathematics in the paper: 80 of 160 questions

Key formulas and results

$\displaystyle\int x^{n}\,dx = \dfrac{x^{n+1}}{n + 1} + C$ , $n \ne -1$
$\displaystyle\int \dfrac{1}{x}\,dx = \log|x| + C$
$\displaystyle\int \dfrac{f'(x)}{f(x)}\,dx = \log|f(x)| + C$
$\displaystyle\int e^{x}[f(x) + f'(x)]\,dx = e^{x} f(x) + C$
$\displaystyle\int \dfrac{dx}{a^{2} + x^{2}} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + C$ ; $\displaystyle\int \dfrac{dx}{\sqrt{a^{2} - x^{2}}} = \sin^{-1}\dfrac{x}{a} + C$
By parts: $\displaystyle\int uv\,dx = u\int v\,dx - \int\left[u'\int v\,dx\right]dx$
$\displaystyle\int \dfrac{dx}{x^{2} - a^{2}} = \dfrac{1}{2a}\log\left|\dfrac{x - a}{x + a}\right| + C$ and $\displaystyle\int \dfrac{dx}{a^{2} - x^{2}} = \dfrac{1}{2a}\log\left|\dfrac{a + x}{a - x}\right| + C$
$\displaystyle\int \dfrac{dx}{\sqrt{x^{2} + a^{2}}} = \log\left|x + \sqrt{x^{2} + a^{2}}\right| + C$ and $\displaystyle\int \dfrac{dx}{\sqrt{x^{2} - a^{2}}} = \log\left|x + \sqrt{x^{2} - a^{2}}\right| + C$
$\displaystyle\int \sqrt{a^{2} - x^{2}}\,dx = \dfrac{x}{2}\sqrt{a^{2} - x^{2}} + \dfrac{a^{2}}{2}\sin^{-1}\dfrac{x}{a} + C$
$\displaystyle\int \sqrt{x^{2} \pm a^{2}}\,dx = \dfrac{x}{2}\sqrt{x^{2} \pm a^{2}} \pm \dfrac{a^{2}}{2}\log\left|x + \sqrt{x^{2} \pm a^{2}}\right| + C$
$\displaystyle\int e^{ax}\sin bx\,dx = \dfrac{e^{ax}(a\sin bx - b\cos bx)}{a^{2} + b^{2}} + C$ and $\displaystyle\int e^{ax}\cos bx\,dx = \dfrac{e^{ax}(a\cos bx + b\sin bx)}{a^{2} + b^{2}} + C$
$\displaystyle\int \sec x\,dx = \log|\sec x + \tan x| + C$ and $\displaystyle\int \operatorname{cosec} x\,dx = \log|\operatorname{cosec} x - \cot x| + C$
Reduction: $\displaystyle\int \tan^{n}x\,dx = \dfrac{\tan^{n-1}x}{n - 1} - \int \tan^{n-2}x\,dx$
Reduction: $\displaystyle\int \sin^{n}x\,dx = -\dfrac{\sin^{n-1}x\cos x}{n} + \dfrac{n - 1}{n}\int \sin^{n-2}x\,dx$

What this chapter covers

Standard integrals
Integration by substitution
Integration by parts
Partial fractions
Integrals of special forms

How it is usually asked

Integrate using a substitution
Integrate by parts
Spot the eˣ[f(x) + f′(x)] pattern
Use partial fractions

Understanding Integration

Integration reverses differentiation: ∫ f(x) dx is a function whose derivative is f(x), plus an arbitrary constant C. The first tool is a table of standard integrals read backwards from the derivative table, including xⁿ, 1/x, eˣ, the trigonometric functions and forms like 1/(a² + x²) and 1/√(a² − x²). Any answer can be checked by differentiating it.

Substitution handles integrands that contain a function and its derivative. If you set t = g(x), then g′(x) dx becomes dt, and the integral becomes a standard one in t. This explains the useful patterns ∫ f′(x)/f(x) dx = log|f(x)| + C and ∫ [f(x)]ⁿ f′(x) dx = [f(x)]ⁿ⁺¹/(n + 1) + C, which can be applied directly once recognised.

Integration by parts, ∫ u v dx = u ∫v dx − ∫ [u′ ∫v dx] dx, handles products such as x eˣ or x sin x; choose u by the ILATE order. A special case is ∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C. Rational functions are split by partial fractions first. EAPCET questions test which method fits and these shortcut patterns.

Worked example

EAPCET-style question

$\displaystyle\int e^{x}\,\dfrac{1 + x\log x}{x}\,dx$ equals

(1) $e^{x}\log x + C$
(2) $\dfrac{e^{x}}{x} + C$
(3) $e^{x}\left(\log x + \dfrac{1}{x}\right) + C$
(4) $xe^{x}\log x + C$

Solution

Split the integrand: $e^{x}\,\dfrac{1 + x\log x}{x} = e^{x}\left[\log x + \dfrac{1}{x}\right]$ .
With $f(x) = \log x$ , $f'(x) = \dfrac{1}{x}$ , so it has the form $e^{x}[f(x) + f'(x)]$ .
Hence the integral is $e^{x} f(x) + C = e^{x}\log x + C$ .
Check: $\dfrac{d}{dx}(e^{x}\log x) = e^{x}\log x + \dfrac{e^{x}}{x}$ , which matches the integrand.

Answer: $e^{x}\log x + C$

Shortcuts and results to remember

Pick u in integration by parts with ILATE: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential.
For $\displaystyle\int \dfrac{dx}{a + b\sin^{2}x}$ or $\displaystyle\int \dfrac{dx}{a\cos^{2}x + b}$ , divide top and bottom by $\cos^{2}x$ and put $\tan x = t$ ; for $\displaystyle\int \dfrac{dx}{a + b\sin x}$ or $(a + b\cos x)$ , use $t = \tan\dfrac{x}{2}$ .
For $\displaystyle\int \dfrac{px + q}{\text{quadratic}}$ , write $px + q = A \times \dfrac{d}{dx}(\text{quadratic}) + B$ and split.
For $\displaystyle\int \dfrac{x^{2} \pm 1}{x^{4} + kx^{2} + 1}\,dx$ , divide by $x^{2}$ and put $t = x \mp \dfrac{1}{x}$ .
In partial fractions an irreducible quadratic factor gets a $(Bx + C)$ numerator, and a repeated factor $(x - a)^{3}$ needs $\dfrac{A}{x - a} + \dfrac{B}{(x - a)^{2}} + \dfrac{C}{(x - a)^{3}}$ .

Common mistakes to avoid

Forgetting to divide by the coefficient of x in a linear inside function; ∫ cos 3x dx = (sin 3x)/3 + C, not sin 3x + C.
Leaving out the 1/a in ∫ dx/(a² + x²); the result is (1/a) tan⁻¹(x/a) + C.
In the eˣ [f + f′] pattern, writing eˣ [f + f′] as the answer; the answer is only eˣ f(x) + C.
Choosing u badly in integration by parts, such as u = eˣ in ∫ x eˣ dx, which makes the integral harder; follow ILATE and take u = x.

How to prepare it

Look first for a function and its derivative in the integrand — f′(x)/f(x) and eˣ[f + f′] patterns are common.

Once the formulas feel familiar, test them the way the exam does: practise Integration questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Integration chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Integration part of the EAPCET Mathematics syllabus?: Yes. Integration is taught in Intermediate second year · Paper IIB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Integration for EAPCET?: Look first for a function and its derivative in the integrand — f′(x)/f(x) and eˣ[f + f′] patterns are common.
How many Mathematics questions are there in EAPCET?: 80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Integration?: Forgetting to divide by the coefficient of x in a linear inside function; ∫ cos 3x dx = (sin 3x)/3 + C, not sin 3x + C. Leaving out the 1/a in ∫ dx/(a² + x²); the result is (1/a) tan⁻¹(x/a) + C. In the eˣ [f + f′] pattern, writing eˣ [f + f′] as the answer; the answer is only eˣ f(x) + C. Choosing u badly in integration by parts, such as u = eˣ in ∫ x eˣ dx, which makes the integral harder; follow ILATE and take u = x.