Question

$(1 \\ 1 + cosx)dx$
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Answer 1

EXPLANATION.

⇒ ∫1/1 + Cos x dx.

As we know that,

Multiply and divide numerator and denominator by (1 - Cos x).

⇒ ∫(1 - Cos x)/(1 + Cos x)(1 - Cos x).

As we know that,

Formula ⇒ (a - b)(a + b) = a² - b².

⇒ ∫(1 - Cos x)/1 - Cos²x dx.

⇒ 1 - Cos²x = Sin²x.

⇒ ∫(1 - Cos x)/Sin²x.

We can write as,

⇒ ∫1/Sin²x dx - ∫Cos x/Sin²x.

⇒ ∫Cosec²x dx - ∫Cot x. Cosec x dx.

⇒ - Cot x - (- Cosec x) + c.

⇒ - Cot x + Cosec x + c.

                                                                                         

MORE INFORMATION.

Standard integrals.

(1) = ∫0.dx = c.

(2) = ∫1.dx = x + c.

(3) = ∫k dx = kx + c, (k ∈ R).

(4) = ∫xⁿdx = xⁿ⁺¹/n + 1 + c, (n ≠ -1).

(5) = ∫dx/x = ㏒(x) + c.

(6) = ∫eˣdx = eˣ + c.

(7) = ∫aˣdx = aˣ/㏒(a) + c = aˣ㏒(e) + c.

Answer 2

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EXPLANATION.

⇒ ∫1/1 + Cos x dx.

As we know that,

Multiply and divide numerator and denominator by (1 - Cos x).

⇒ ∫(1 - Cos x)/(1 + Cos x)(1 - Cos x).

As we know that,

Formula ⇒ (a - b)(a + b) = a² - b².

⇒ ∫(1 - Cos x)/1 - Cos²x dx.

⇒ 1 - Cos²x = Sin²x.

⇒ ∫(1 - Cos x)/Sin²x.

We can write as,

⇒ ∫1/Sin²x dx - ∫Cos x/Sin²x.

⇒ ∫Cosec²x dx - ∫Cot x. Cosec x dx.

⇒ - Cot x - (- Cosec x) + c.

⇒ - Cot x + Cosec x + c.

                                                                                         

MORE INFORMATION.

Standard integrals.

(1) = ∫0.dx = c.

(2) = ∫1.dx = x + c.

(3) = ∫k dx = kx + c, (k ∈ R).

(4) = ∫xⁿdx = xⁿ⁺¹/n + 1 + c, (n ≠ -1).

(5) = ∫dx/x = ㏒(x) + c.

(6) = ∫eˣdx = eˣ + c.

(7) = ∫aˣdx = aˣ/㏒(a) + c = aˣ㏒(e) + c.