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

Studdy Solution

STEP 1

Assumptions1. The integral is a definite integral. . We are integrating the function $\frac{\cos (t)}{\sin ^{}(t)}$ with respect to $t$ .

STEP 2

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

STEP 3

Now, we need to find the derivative of $u$ with respect to $t$ , which we denote as $du/dt$ or $u'$ .
$u' = \cos(t)$

STEP 4

We can rewrite $du/dt$ as $du = \cos(t) dt$ .
$du = \cos(t) dt$

STEP 5

Now, we can substitute $u$ and $du$ into the integral.
$\int \frac{\cos (t)}{\sin ^{2}(t)} dt = \int \frac{1}{u^{2}} du$

STEP 6

This integral is now in a form that we can easily solve. The integral of $\frac{1}{u^{2}}$ is $-\frac{1}{u}$ .
$\int \frac{1}{u^{2}} du = -\frac{1}{u} + C$

STEP 7

Now, we substitute $u$ back in terms of $t$ .
$-\frac{1}{u} + C = -\frac{1}{\sin(t)} + C$ So, the solution to the integral $\int \frac{\cos (t)}{\sin ^{2}(t)} dt$ is $-\frac{1}{\sin(t)} + C$ .

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

Studdy Solution

STEP 1

Assumptions1. The integral is a definite integral. . We are integrating the function $\frac{\cos (t)}{\sin ^{}(t)}$ with respect to $t$ .

STEP 2

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

STEP 3

Now, we need to find the derivative of $u$ with respect to $t$ , which we denote as $du/dt$ or $u'$ .
$u' = \cos(t)$

STEP 4

We can rewrite $du/dt$ as $du = \cos(t) dt$ .
$du = \cos(t) dt$

STEP 5

Now, we can substitute $u$ and $du$ into the integral.
$\int \frac{\cos (t)}{\sin ^{2}(t)} dt = \int \frac{1}{u^{2}} du$

STEP 6

This integral is now in a form that we can easily solve. The integral of $\frac{1}{u^{2}}$ is $-\frac{1}{u}$ .
$\int \frac{1}{u^{2}} du = -\frac{1}{u} + C$

STEP 7

Now, we substitute $u$ back in terms of $t$ .
$-\frac{1}{u} + C = -\frac{1}{\sin(t)} + C$ So, the solution to the integral $\int \frac{\cos (t)}{\sin ^{2}(t)} dt$ is $-\frac{1}{\sin(t)} + C$ .

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QuestionFind the integral of cos⁡(t)sin⁡2(t)\frac{\cos (t)}{\sin ^{2}(t)}sin2(t)cos(t)​ with respect to ttt.

Studdy Solution

STEP 1

Assumptions1. The integral is a definite integral. . We are integrating the function cos⁡(t)sin⁡(t)\frac{\cos (t)}{\sin ^{}(t)}sin(t)cos(t)​ with respect to ttt.

STEP 2

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

STEP 3

Now, we need to find the derivative of uuu with respect to ttt, which we denote as du/dtdu/dtdu/dt or u′u'u′.u′=cos⁡(t)u' = \cos(t)u′=cos(t)

STEP 4

We can rewrite du/dtdu/dtdu/dt as du=cos⁡(t)dtdu = \cos(t) dtdu=cos(t)dt.du=cos⁡(t)dtdu = \cos(t) dtdu=cos(t)dt

STEP 5

Now, we can substitute uuu and dududu into the integral.∫cos⁡(t)sin⁡2(t)dt=∫1u2du\int \frac{\cos (t)}{\sin ^{2}(t)} dt = \int \frac{1}{u^{2}} du∫sin2(t)cos(t)​dt=∫u21​du

STEP 6

This integral is now in a form that we can easily solve. The integral of 1u2\frac{1}{u^{2}}u21​ is −1u-\frac{1}{u}−u1​.∫1u2du=−1u+C\int \frac{1}{u^{2}} du = -\frac{1}{u} + C∫u21​du=−u1​+C

STEP 7

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

Assumptions1. The integral is a definite integral. . We are integrating the function $\frac{\cos (t)}{\sin ^{}(t)}$ with respect to $t$ .

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

Now, we need to find the derivative of $u$ with respect to $t$ , which we denote as $du/dt$ or $u'$ .
$u' = \cos(t)$

We can rewrite $du/dt$ as $du = \cos(t) dt$ .
$du = \cos(t) dt$

Now, we can substitute $u$ and $du$ into the integral.
$\int \frac{\cos (t)}{\sin ^{2}(t)} dt = \int \frac{1}{u^{2}} du$

This integral is now in a form that we can easily solve. The integral of $\frac{1}{u^{2}}$ is $-\frac{1}{u}$ .
$\int \frac{1}{u^{2}} du = -\frac{1}{u} + C$