What is the integral of sin(x)(1+sec^2(x)) ?

The integral of sin(x)(1+sec^2(x)) is sin(x)tan(x)+C

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integral of sin(x)(1+sec^2(x))

\int sin (x) (1 + sec^{2} (x)) d x

sin (x) tan (x) + C

Solution steps

\int sin (x) (1 + sec^{2} (x)) d x

Apply u-substitution

: \int \frac{1 + u ^{2}}{u ^{2}} d u

= \int \frac{1 + u ^{2}}{u ^{2}} d u

\frac{1 + u ^{2}}{u ^{2}} = \frac{1}{u ^{2}} + 1

= \int \frac{1}{u ^{2}} + 1 d u

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{u ^{2}} = u^{- 2}

= \int u^{- 2} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{- 2} d u + \int 1 d u

\int u^{- 2} d u = - \frac{1}{u}

\int 1 d u = u

= - \frac{1}{u} + u

Substitute back

u = sec (x)

= - \frac{1}{sec ( x )} + sec (x)

Simplify

- \frac{1}{sec ( x )} + sec (x) : sin (x) tan (x)

= sin (x) tan (x)

Add a constant to the solution

= sin (x) tan (x) + C

\frac{d x}{d t} = 1.5 t cos^{2} (x), x (0) = - 1.4

f (x) = 4 x - 4 x

n = 2 \sum \infty \frac{7}{6 ^{n}}

\int 23 arctan (x) d x

f (x) = cos (5 x^{2} + 5)

What is the integral of sin(x)(1+sec^2(x)) ?
The integral of sin(x)(1+sec^2(x)) is sin(x)tan(x)+C