Math

QuestionFind the integral of sin(t)cos2(t)\frac{\sin (t)}{\cos ^{2}(t)} with respect to tt.

Studdy Solution

STEP 1

Assumptions1. We are given the integral sin(t)cos(t)dt\int \frac{\sin (t)}{\cos ^{}(t)} d t . We are familiar with the basic rules of integration and trigonometric identities

STEP 2

We can simplify the integral by using a substitution method. Let's set u=cos(t)u = \cos(t).
u=cos(t)u = \cos(t)

STEP 3

Differentiate uu with respect to tt to get du/dtdu/dt.
dudt=sin(t)\frac{du}{dt} = -\sin(t)

STEP 4

Rearrange the equation to get dtdt in terms of dudu.
dt=duudt = -\frac{du}{u'}

STEP 5

Substitute the values of uu and dtdt into the integral.
sin(t)cos2(t)dt=1u2du\int \frac{\sin (t)}{\cos ^{2}(t)} d t = \int \frac{1}{u^{2}} \cdot -du

STEP 6

implify the integral.
1u2du=u2du\int \frac{1}{u^{2}} \cdot -du = -\int u^{-2} du

STEP 7

Now, we can integrate the function. The integral of u2u^{-2} is u1-u^{-1}.
u2du=(u1)-\int u^{-2} du = -(-u^{-1})

STEP 8

implify the result.
(u1)=u1-(-u^{-1}) = u^{-1}

STEP 9

Substitute back the original value of uu which is cos(t)\cos(t).
u=(cos(t))u^{-} = (\cos(t))^{-}

STEP 10

implify the result.
(cos(t))=sec(t)(\cos(t))^{-} = \sec(t)So, the integral of sin(t)cos2(t)\frac{\sin (t)}{\cos ^{2}(t)} with respect to tt is sec(t)\sec(t).

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