INTEGRATION
INTEGRATION
✤ FORMULA
1. ∫xⁿdx = xn+1 / (n+1) + C.
2. ∫eⁿ dx = eˣ + C
3. ∫aⁿ dx = aˣ / log a + C.
4. ∫1/x dx = log x + c:
5. ∫sinx dx = -cos x + C
6. ∫cosx dx = sin x + C.
7. ∫secx dx = tanx +C
8. ∫cosec²x dx = -cotx + C
9. ∫tanx dx = log Secx = -log(cosx)+C
10. ∫cotx dx = log sin x + c.
11. ∫ Secn dx = log | secx + tan x| / c.
12. ∫cosec x dx = log |cosec x - cotn/c.
13. ∫dx = x + c.
✤ Algebra of integration.
1. addition Rule
∫(f(x) + g(x) dx = ∫f(x) dx + ∫g(x)de
2. Substraction Rule
∫(f(x) - g(x)) dx = ∫f(x)da - ∫g(x)de
3. Scalar Rule
∫k⋅f(x)dx = k⋅∫f(x)dx.
4. Prodxct Rule
∫f(x)g(x) dx = f(x)∫g(x) dx - ∫(df(x)/dx) ∫g(x) from
Some New formula.
1. ∫ dx/ a²+x² = 1/a tax⁻¹(x/a) + C
2. ∫ dx/ a²-x² = 1/2a log |a+x/ a-x| + C
3. ∫ dx/ x²-a² = 1/2a log |x-a/x+a| + C
4. ∫ dx/ √a²-x² = six⁻¹(x/a) + c
5. ∫ dx/ √x²-a² = log |x+√x²-a²| + c
6. ∫ dx/√a²+x² = log |x+√a²+x²| + c.
Note: To apply these formula, coefficient of n² should be one.
7. ∫ dx/√1-x² = six⁻¹x + c
8. ∫ dx/ 1+x² = tax⁻¹x+c
9. ∫ dx/ x√x²-1 = cosec⁻¹x + c.
IMPORTANT POINTS TO REMEMBER
1. Steps to solve the Questions of
∫ 1/ax²+bx+c.
i) divide and multiply by coff. of x².
ii) add and substract the square of half of coeff of x
iii) adjust the equation in the form of
1/a²-m or 1/a²+x or 1/x²-a² and
apply the formula...
2. Steps to solve ∫dx/√ax²+bx+c
i) Apply the same concept of above.
ii) after adjusted, apply the formula
1/√a²-x² or 1/√a²+x² or 1/√x²-a²
3. ∫ Px+q dx/√ax²+bx+c
steps:
i) Let Px+q = A d/dn (ax²+bx+c)+B.
ii) find A and B in the terms of P and q. (A = P/2a, B = ..)
iii) Now adjust the equation as
∫ A(2ax+b)dx/√ax²+bx+c + ∫ B/√ax²+bx+c
= A log (ax²+bx+c) + B . Apply the above concept.
4: ∫ Px+q dx/√ax²+bx+c
i) Apply
Px+q= A d/dx (ax²+bx+c)+B
ii) find A and B as in type3
iii) At last
integral is:
2A √ax²+bx+c + B ∫ dx/√ax²+bx+c.
Types: Partial fraction.
-some standard results
1. 1/(x+a)(x+b) = A/(x+a) + B/(x+b)
2. 1/(x+a)²(x+b) = A/(x+a) + B/(x+a)² + C/(x+b)
3. 1/(x²+a) (x+b) = Ax+B/(x²+a) + C/(x+b)
4. P(x)/Q(x); if degree of P(x)≥ Q(x)
then divide P(x) by Q(x) and
write Q + R/Q(x) → Remainder
↓
Quotient
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