Integrate the following
(i) sec2 (3x + 7)
Answers
EXPLANATION.
⇒ ∫sec²(3x + 7)dx.
As we know that,
By using the substitution method, we get.
Let we assume that,
⇒ 3x + 7 = t.
Differentiate w.r.t x, we get.
⇒ 3dx = dt.
⇒ dx = dt/3.
Put the value in the equation, we get.
⇒ ∫sec²(t)dt/3.
⇒ 1/3∫sec²tdt.
⇒ 1/3 tan(t) + C.
Put the value of t = 3x + 7 in the equation, we get.
⇒ 1/3 tan(3x + 7) + C.
MORE INFORMATION.
Standard integrals.
(1) = ∫sin x dx = - cos x + c.
(2) = ∫cos x dx = sin x + c.
(3) = ∫tan x dx = ㏒(sec x) + c.
(4) = ∫cot x dx = ㏒(sin x) + c.
(5) = ∫sec x dx = ㏒(sec x + tan x) + c. = - ㏒(sec x - tan x) + c. = ㏒ tan(π/4 + x/2) + c.
(6) = ∫cosec x dx = - ㏒(cosec x + cot x) + c. = ㏒(cosec x - cot x) + c = ㏒ tan(x/2) + c.
(7) = ∫sec x tan x dx = sec x + c.
(8) = ∫cosec x cot x dx = - cosec x + c.
(9) = ∫sec²xdx = tan x + c.
(10) = ∫cosec²xdx = - cot x + c.
⇒ ∫sec²(3x + 7)dx.
As we know that,
By using the substitution method, we get.
Let we assume that,
⇒ 3x + 7 = t.
Differentiate w.r.t x, we get.
⇒ 3dx = dt.
⇒ dx = dt/3.
Put the value in the equation, we get.
⇒ ∫sec²(t)dt/3.
⇒ 1/3∫sec²tdt.
⇒ 1/3 tan(t) + C.
Put the value of t = 3x + 7 in the equation, we get.
⇒ 1/3 tan(3x + 7) + C.