Integration Formulas for Class 12 – Indefinite Integration

Here you will learn Integration integration formulas for class 12 and integration by substitution with example.

Let’s begin –

Integration Formula for Class 12

(i)  \(\int\) \({(ax+b)}^n\) dx = \({(ax+b)}^{n+1}\over {a(n+1)}\) + C ; n \(\ne\) -1

(ii)  \(\int\) \(dx\over {ax+b}\) dx = \(1\over a\) ln|ax+b| + C

(iii)  \(\int\) \(e^{ax+b}\) dx = \({1\over {a}}e^{ax+b}\) + C or \(\int\) \(e^x\) = \(e^x\) + C

(iv)  \(\int\) \(a^{px+q}\) dx = \({1\over p}\) \({a^{px+q}}\over lna\) + C, (a > 0)

(v)  \(\int\) sinx dx = -cosx + C

(vi)  \(\int\) cosx dx = sinx + C

(vii)   \(\int\) tanx dx = ln|secx| + C

(viii)  \(\int\) cotx dx = ln|sinx| + C

(ix)  \(\int\) \(sec^2x\) dx = tanx + C

(x)  \(\int\) \(cosec^2x\) dx = -cotx + C

(xi)  \(\int\) cosecx.cotx dx = -cosecx + C

(xii)  \(\int\) secx.tanx dx = secx + C

(xiii)  \(\int\) secx dx = ln|secx+tanx| + C = ln|tan(\(\pi\over 4\) + \(x\over 2\))| + C

(xiv)  \(\int\) cosecx dx = ln|cosecx-cotx| + C = ln|tan\(x\over 2\)| = -ln|cosecx+cotx| + C

(xv)  \(\int\) \(dx\over {\sqrt{a^2-x^2}}\) = \(sin^{-1} {x\over a}\) + C

(xvi)  \(\int\) \(dx\over {a^2+x^2}\) = \(1\over a\) \(tan^{-1} {x\over a}\) + C

(xvii)  \(\int\) \(dx\over {x\sqrt{x^2-a^2}}\) = \(1\over a\) \(sec^{-1} {x\over a}\) + C

(xviii) \(\int\) \(dx\over {\sqrt{x^2+a^2}}\) = \(ln[x+\sqrt{x^2+a^2}]\) + C

(xix)  \(\int\) \(dx\over {\sqrt{x^2-a^2}}\) = \(ln[x+\sqrt{x^2-a^2}]\) + C

(xx)  \(\int\) \(dx\over {a^2-x^2}\) = \(1\over 2a\) \(ln|{a+x\over {a-x}}|\) + C

(xxi)  \(\int\) \(dx\over {x^2-a^2}\) = \(1\over 2a\) \(ln|{x-a\over {x+a}}|\) + C

(xxii)  \(\int\) \(\sqrt{a^2-x^2}\) dx = \(x\over 2\)\(\sqrt{a^2-x^2}\) + \(a^2\over 2\) \(sin^{-1} {x\over a}\) + C

(xxii)  \(\int\) \(\sqrt{x^2+a^2}\) dx = \(x\over 2\)\(\sqrt{x^2+a^2}\) + \(a^2\over 2\) \(ln[x+\sqrt{x^2+a^2}]\) + C

(xxii)  \(\int\) \(\sqrt{x^2-a^2}\) dx = \(x\over 2\)\(\sqrt{x^2-a^2}\) – \(a^2\over 2\) \(ln[x+\sqrt{x^2-a^2}]\) + C

Integration by Substitution


Example : Evaluate \(\int\) \(cos^2x\over {sin^2x + sinx}\) dx

Solution : I = \(\int\) \((1-sin^2x)cosx\over {sinx(1 + sinx)}\) dx = \(\int\) \(1 – sinx\over {sinx}\) cosx dx

Put sinx = t   \(\implies\)   cosx dx = dt

\(\implies\) I = \(\int\) \(1-t\over t\) dt = ln|t| – t + C = ln|sinx| – sinx + C

Hope you learnt integration formulas for class 12, learn more concepts of integration and practice more questions to get ahead in competition. Good Luck!

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