integral (2tan(x))/((1+2tan^2(x)))

الحلّ

integral

\frac{2 tan ( x )}{( 1 + 2 tan ^{2} ( x ) )}

ln 1 + sin^{2} (x) + C

خطوات الحلّ

\int \frac{2 tan ( x )}{1 + 2 tan ^{2} ( x )} d x

\int a \cdot f (x) d x = a \cdot \int f (x) d x

:قم بإخراج الثابت

= 2 \cdot \int \frac{tan ( x )}{1 + 2 tan ^{2} ( x )} d x

Rewrite using trig identities

= 2 \cdot \int \frac{sin ( x ) cos ( x )}{cos ^{2} ( x ) + 2 sin ^{2} ( x )} d x

Rewrite using trig identities

= 2 \cdot \int \frac{sin ( x ) cos ( x )}{1 + sin ^{2} ( x )} d x

Rewrite using trig identities

= 2 \cdot \int \frac{sin ( 2 x )}{2 ( 1 + sin ^{2} ( x ) )} d x

\int a \cdot f (x) d x = a \cdot \int f (x) d x

:قم بإخراج الثابت

= 2 \cdot \frac{1}{2} \cdot \int \frac{sin ( 2 x )}{1 + sin ^{2} ( x )} d x

فعّل طريقة التكامل بالتعويض

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

\int \frac{1}{u} d u = ln (∣ u ∣)

:استخدم التكامل البسيط

= 2 \cdot \frac{1}{2} ln ∣ u ∣

u = 1 + sin^{2} (x)

استبدل مجددًا

= 2 \cdot \frac{1}{2} ln 1 + sin^{2} (x)

2 \cdot \frac{1}{2} ln 1 + sin^{2} (x)

بسّط:

ln 1 + sin^{2} (x)

= ln 1 + sin^{2} (x)

أضف ثابت التكامل للحل

= ln 1 + sin^{2} (x) + C

in t e g r a l \frac{( x + 1 ) ^{2}}{( x - 1 ) ^{2}}

in t e g r a l tan (2 - 3 x)

in t e g r a l e^{x^{2}} x^{2}

in t e g r a l 3 x^{2} - 9 x + 5

in t e g r a l \frac{e ^{x}}{3}