Question

integrate this function​

Answer 1

EXPLANATION.

⇒ ∫(eˣ - sin x)/(eˣ + cos x) dx.

As we know that,

We can using substitution method in this equation, we get.

Let we assume that,

⇒ eˣ + cos x = t.

Differentiate w.r.t x, we get.

⇒ (eˣ - sin x)dx = dt.

Put the values in the equation, we get.

⇒ ∫dt/t.

⇒ ㏑|t| + c.

Put the value of t = eˣ + cos x in the equation, we get.

⇒ ㏑|eˣ + cos x| + c.

                                                                                                                       

MORE INFORMATION.

Integration by parts.

(1) = If u and v are 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 functions.

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) + c.

(3) = If the integral is of the form,

⇒ ∫[xf'(x) + f(x)]dx = x f(x) + c.