integrate tanx sec³x dx
Answers
Step-by-step explanation:
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EXPLANATION.
⇒ ∫tan(x).sec³xdx.
As we know that,
We can write as,
⇒ ∫tan(x).sec(x).sec²xdx.
By using substitution method, we get.
Let,
⇒ Sec(x) = t.
Differentiate w.r.t x we get,
Sec(x).tan(x)dx = dt.
⇒ ∫t²dt.
⇒ t³/3 + c.
Put the value of t in equation, we get.
⇒ Sec³x/3 + c.
MORE INFORMATION.
Integration by parts.
(1) = If u and v are the two functions of x then,
∫(u. v)dx = u(∫v dx) - ∫[(du/dx).(∫v dx)dx].
From the first letter of the words,
I = inverse trigonometric functions.
L = logarithmic functions.
A = algebraic function.
T = trigonometric functions.
E = exponential functions.
We get a word = ILATE.
Therefore, first arrange the functions in the order according to letters of this word and then integrate by parts.
(2) = if the integral is of the form ∫eˣ [f(x) + f'(x)]dx.
∫eˣ [f(x) + f'(x)]dx = eˣf(x) + c.
(3) = If the integral is of the form ∫ [xf'(x) + f(x)]dx.
∫ [xf'(x) + f(x)]dx = x f(x) + c.