Calculus Examples

$\int_{0}^{\frac{π}{6}} \frac{\sin (t)}{\cos^{2} (t)} d t$

Step 1

Let $u = \cos (t)$ . Then $d u = - \sin (t) d t$ , so $- d u = \sin (t) d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \cos (t)$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $\cos (t)$ .

$\frac{d}{d t} [\cos (t)]$

Step 1.1.2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$- \sin (t)$

Step 1.2

Substitute the lower limit in for $t$ in $u = \cos (t)$ .

$u_{lower} = \cos (0)$

Step 1.3

The exact value of $\cos (0)$ is $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $t$ in $u = \cos (t)$ .

$u_{upper} = \cos (\frac{π}{6})$

Step 1.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$u_{upper} = \frac{\sqrt{3}}{2}$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = \frac{\sqrt{3}}{2}$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{\frac{\sqrt{3}}{2}} \frac{- 1}{u^{2}} d u$

Step 2

Move the negative in front of the fraction.

$\int_{1}^{\frac{\sqrt{3}}{2}} - \frac{1}{u^{2}} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{1}^{\frac{\sqrt{3}}{2}} \frac{1}{u^{2}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

$- \int_{1}^{\frac{\sqrt{3}}{2}} {(u^{2})}^{- 1} d u$

Step 4.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \int_{1}^{\frac{\sqrt{3}}{2}} u^{2 \cdot - 1} d u$

Step 4.2.2

Multiply $2$ by $- 1$ .

$- \int_{1}^{\frac{\sqrt{3}}{2}} u^{- 2} d u$

Step 5

By the Power Rule, the integral of $u^{- 2}$ with respect to $u$ is $- u^{- 1}$ .

$- (- u^{- 1}]_{1}^{\frac{\sqrt{3}}{2}})$

Step 6

Evaluate $- u^{- 1}$ at $\frac{\sqrt{3}}{2}$ and at $1$ .

$- ((- {(\frac{\sqrt{3}}{2})}^{- 1}) + 1^{- 1})$

Step 7

Change the sign of the exponent by rewriting the base as its reciprocal.

$- (- \frac{2}{\sqrt{3}} + 1^{- 1})$

Step 8

One to any power is one.

$- (- \frac{2}{\sqrt{3}} + 1)$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- (- \frac{2}{\sqrt{3}} + 1)$

Decimal Form:

$0.15470053 \dots$