PYQ Vault
EbooksTutor

CBSE Class 12 Maths

Integrals

Each concept is a short lesson, then questions until you get three right.

0/17

17 concepts, a short lesson each

  • Antiderivatives0/2
  • Substitution0/2
  • Quadratic forms0/3
  • Parts and partial fractions0/3
  • Evaluating0/3
  • Properties0/3
  • Beyond the book0/1

Start here

Integration as the reverse of differentiation

∫f(x) dx=F(x)+c\int f(x)\,dx = F(x) + c when F′(x)=f(x)F'(x) = f(x); use the standard formulae term by term, e.g. ∫xn dx=xn+1n+1+c\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c for n≠−1n \ne -1

Start

Antiderivatives0 of 2

  1. Integration as the reverse of differentiation∫f(x) dx=F(x)+c\int f(x)\,dx = F(x) + c when F′(x)=f(x)F'(x) = f(x); use the standard formulae term by term, e.g. ∫xn dx=xn+1n+1+c\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + c for n≠−1n \ne -1Start
  2. Rewriting before integratingExpand, divide out or use an identity first (sin⁡2x=1−cos⁡2x2\sin^2 x = \frac{1 - \cos 2x}{2}, tan⁡2x=sec⁡2x−1\tan^2 x = \sec^2 x - 1) so every term matches a standard formStart

Substitution0 of 2

  1. Integration by substitutionPut u=g(x)u = g(x) so that g′(x) dx=dug'(x)\,dx = du: ∫f(g(x)) g′(x) dx=∫f(u) du\int f(g(x))\,g'(x)\,dx = \int f(u)\,du; in particular ∫f′(x)f(x) dx=log⁡∣f(x)∣+c\int \dfrac{f'(x)}{f(x)}\,dx = \log|f(x)| + cStart
  2. Powers and products of trigonometric functionsTurn products into sums (2cos⁡Acos⁡B=cos⁡(A+B)+cos⁡(A−B)2\cos A\cos B = \cos(A + B) + \cos(A - B)); for sin⁡mxcos⁡nx\sin^m x\cos^n x with an odd power, keep one factor and substitute for the other functionStart

Quadratic forms0 of 3

  1. Standard integrals of quadratic forms∫dxx2+a2=1atan⁡−1xa\int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a}, ∫dxx2−a2=12alog⁡∣x−ax+a∣\int \dfrac{dx}{x^2 - a^2} = \dfrac{1}{2a}\log\left|\dfrac{x - a}{x + a}\right|, ∫dxa2−x2=sin⁡−1xa\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\dfrac{x}{a}, and the rest of the tableStart
  2. Completing the squareWrite ax2+bx+cax^2 + bx + c as a[(x+p)2±q2]a\left[(x + p)^2 \pm q^2\right], then use the standard form with x+px + p in place of xxStart
  3. Integrals of (px + q) over a quadraticWrite px+q=A ddx(ax2+bx+c)+Bpx + q = A\,\dfrac{d}{dx}(ax^2 + bx + c) + B; the AA part gives a log or a root, the BB part a standard quadratic formStart

Parts and partial fractions0 of 3

  1. Integration by parts∫u v dx=u∫v dx−∫(u′∫v dx)dx\int u\,v\,dx = u\int v\,dx - \int \left(u'\int v\,dx\right)dx; choose uu by LIATE (log, inverse trig, algebraic, trig, exponential)Start
  2. Special forms from integration by parts∫ex[f(x)+f′(x)]dx=exf(x)+c\int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + c; ∫a2−x2 dx\int \sqrt{a^2 - x^2}\,dx and its relatives; integrals that return to themselves, like ∫exsin⁡x dx\int e^x\sin x\,dxStart
  3. Integration by partial fractionsSplit a proper rational function into Ax−a+Bx−b+…\dfrac{A}{x - a} + \dfrac{B}{x - b} + \dots (repeated and quadratic factors too), then integrate each pieceStart

Evaluating0 of 3

  1. The fundamental theorem of integral calculusIf F′(x)=f(x)F'(x) = f(x), then ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a): find any antiderivative and subtract its values at the limitsStart
  2. Substitution in a definite integralWhen you put u=g(x)u = g(x), change the limits too: x=ax = a becomes u=g(a)u = g(a), x=bx = b becomes u=g(b)u = g(b); then never go back to xxStart
  3. Parts and partial fractions with limits∫abu v dx=[u∫v dx]ab−∫ab(u′∫v dx)dx\int_a^b u\,v\,dx = \left[u\int v\,dx\right]_a^b - \int_a^b \left(u'\int v\,dx\right)dx; split rational functions first, then evaluateStart

Properties0 of 3

  1. Basic properties and splitting the interval∫ab=−∫ba\int_a^b = -\int_b^a, ∫ab=∫ac+∫cb\int_a^b = \int_a^c + \int_c^b; split at the point where ∣x−c∣|x - c| or a piecewise function changes formStart
  2. The property f(a + b − x)∫abf(x) dx=∫abf(a+b−x) dx\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx; add the two forms when they sum to something simple, as in ∫0π/2sin⁡xsin⁡x+cos⁡x dx=π4\int_0^{\pi/2}\dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx = \dfrac{\pi}{4}Start
  3. Odd and even functions, and the 2a property∫−aaf(x) dx\int_{-a}^{a} f(x)\,dx is 00 for odd ff and 2∫0af(x) dx2\int_0^a f(x)\,dx for even ff; ∫02af(x) dx=2∫0af(x) dx\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx when f(2a−x)=f(x)f(2a - x) = f(x)Start

Beyond the book0 of 1

  1. Integrands with [x] and |x|Split the interval where [x][x] jumps or the expression inside ∣ ∣|\ | changes sign; on each piece the integrand is a plain functionStart

Practise Integrals

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions