Integral Calculus

∫xndx=xn+1n+1+C,n≠−1

∫cosxdx=sinx+C

∫sinxdx=−cosx+C

∫sec2xdx=tanx+C

∫cosec2xdx=−cotx+C

∫secxtanxdx=secx+C

∫cosecxcotxdx=−cosecx+C

∫dx1−x2√=sin−1x+C

∫dx1−x2√=−cos−1x+C

∫dx1+x2=tan−1x+C

∫dx1+x2=−cot−1x+C

∫exdx=ex+C

∫axdx=axloga+C

∫dxxx2−1√=sec−1x+C

∫dxxx2−1√=−cosec−1x+C

∫dxxx2−1√=−cosec−1x+C

∫1xdx=log|x|+C

Integration by Substitution Formulas

• 1. ∫tanxdx=log|secx|+C
  
• 2. ∫cotxdx=log|sinx|+C
  
• 3. ∫secxdx=log|secx+tanx|+C
  
• 4. ∫cosecxdx=log|cosecx−cotx|+C

Integrals (Special Functions) Formulas

• i) ∫dxx2−a2=12alog∣∣x−ax+a∣∣+C
  
• ii) ∫dxa2−x2=12alog∣∣a+xa−x∣∣+C
• iii) ∫dxx2+a2=1atan−1xa+C
  
• iv) ∫dxx2−a2√=log∣∣x+x2−a2−−−−−−√∣∣+C
  
• v) ∫dxx2+a2√=log∣∣x+x2+a2−−−−−−√∣∣+C
  
• vi) ∫dxx2−a2√=sin−1xa+C

Integration by Parts Formulas

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• ∫f1(x).f2(x)=f1(x)∫f2(x)dx−∫[ddxf1(x).∫f2(x)dx]dx
  
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• ∫ex[f(x)+f′(x)]dx=∫exf(x)dx+C

Special Integrals Formulas

a) ∫x2−a2−−−−−−√dx=x2x2−a2−−−−−−√−a22log∣∣x+x2−a2−−−−−−√∣∣+C

• b) ∫x2+a2−−−−−−√dx=x2x2+a2−−−−−−√+a22log∣∣x+x2+a2−−−−−−√∣∣+C
  
• c) ∫a2−x2−−−−−−√dx=x2a2−x2−−−−−−√+a2sin−1xa+C
  
• d) ax2+bx+c=a[x2+bax+ca]=a[(x+b2a)2+(ca−b24a2)]