find the value of x so ,that (2/7)²^x×(2/7)^x
=(2/7)⁶
Answers
Answer:
Example 1:
Integrate x⁶ dx
Solution:
Formula :
∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c
∫x⁶ dx = x⁽⁶ ⁺ ¹⁾ /(6 + 1) + c
= x⁷/7 + c
Example 2:
Integrate x ⁻² dx
Solution:
Formula :
∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c
∫x⁶ dx = x⁽⁻² ⁺ ¹⁾ /(-2 + 1) + c
= x⁻¹/(-1) + c
= (-1/x) + c
Example 3:
Integrate √x⁵ dx
Solution:
Formula :
∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c
∫√x⁵ dx = ∫ (x⁵)^1/2 dx
= ∫ (x^5/2) dx
= x^[(5/2) + 1)]/[(5/2) + 1)] + C
= x^[(5 + 2)/2)]/[(5 + 2)/2)] + C
= x^(7/2)/(7/2) + C
= (2/7)x^(7/2) + C simple problems on integration
Example 4:
Integrate Sin x/Cos ² x dx
Solution:
∫Sin x/Cos ² x dx = ∫(Sin x/Cos x) x (1/ Cos x) dx
= ∫ tan x sec x dx
Formula :
∫ sec x tan x dx = sec x + c
= Sec x + c
Example 5:
Integrate Cot x/Sin x dx
Solution:
∫ (Cot x/Sin x) dx = ∫(cot x) x (1/ sin x) dx
= ∫ Cot x Cosec x dx
Formula :
∫ Cosec x cot x dx = - Cosec x + c
= - Cosec x + c
Example 6:
Integrate 1/Sin² x dx
Solution:
∫ 1/Sin ²x dx = ∫ Cosec ² x dx
Formula :
∫ Cosec ² x dx = - Cot x + c
= - Cot x + c
Example 7:
Integrate (3-4x)⁶ dx
Solution:
Formula :
∫ (ax + b)ⁿ dx = (1/a) (ax + b)⁽ⁿ ⁺ ¹⁾/(n + 1) + c
∫ (3-4x)⁶ dx = (3-4x)⁽⁶ ⁺ ¹⁾/(6 + 1) (-1/4) + C
= (-1/4) (3-4x)⁷/7 + C
= (-7/4) (3-4x)⁷ + C
Example 8:
Integrate 1/(3+5x) dx
Solution:
Formula :
∫ 1/(ax + b) dx = (1/a) log (ax + b) + c
∫ 1/(3+5x) dx = (1/5) log (3+5x) + C
Example 9:
Integrate Cosec (4x + 3) cot (4x + 3) dx
Solution:
Formula :
∫ Cosec (ax+b)cot (ax+b)dx=-(1/a)Cosec (ax+b) + c
∫ Cosec (4x + 3) cot (4x + 3) dx = - Cosec (4x + 3) (1/4) + C
= - (1/4) Cosec (4x + 3) + C
Example 10:
Integrate Sec (ax + b) tan (ax + b) dx
Solution:
Formula :
∫ sec (ax + b) tan (ax + b) dx = sec (ax + b) + c
∫ Sec (ax + b) tan (ax + b) dx = Sec (ax + b) (1/a) + C
= (1/a) Sec (ax + b) (1/a) + C
These are the example problems in the topic integration.