Calculus Examples

$\int_{0}^{\frac{π}{4}} \frac{\sin (t)}{\cos^{2} (t)} d t$

Step 1

Multiply by $1$ .

$\int_{0}^{\frac{π}{4}} \frac{\sin (t) \cdot 1}{\cos^{2} (t)} d t$

Step 2

Factor $\cos (t)$ out of $\cos^{2} (t)$ .

$\int_{0}^{\frac{π}{4}} \frac{\sin (t) \cdot 1}{\cos (t) \cos (t)} d t$

Step 3

Separate fractions.

$\int_{0}^{\frac{π}{4}} \frac{\sin (t)}{\cos (t)} \cdot \frac{1}{\cos (t)} d t$

Step 4

Convert from $\frac{\sin (t)}{\cos (t)}$ to $\tan (t)$ .

$\int_{0}^{\frac{π}{4}} \tan (t) \frac{1}{\cos (t)} d t$

Step 5

Convert from $\frac{1}{\cos (t)}$ to $\sec (t)$ .

$\int_{0}^{\frac{π}{4}} \tan (t) \sec (t) d t$

Step 6

Since the derivative of $\sec (t)$ is $\tan (t) \sec (t)$ , the integral of $\tan (t) \sec (t)$ is $\sec (t)$ .

$\sec (t)]_{0}^{\frac{π}{4}}$

Step 7

Simplify the answer.

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Step 7.1

Evaluate $\sec (t)$ at $\frac{π}{4}$ and at $0$ .

$\sec (\frac{π}{4}) - \sec (0)$

Step 7.2

Simplify.

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Step 7.2.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{2}{\sqrt{2}} - \sec (0)$

Step 7.2.2

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

$\frac{2}{\sqrt{2}} - 1 \cdot 1$

Step 7.2.3

Multiply $- 1$ by $1$ .

$\frac{2}{\sqrt{2}} - 1$

Step 7.3

Simplify each term.

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Step 7.3.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} - 1$

Step 7.3.2

Combine and simplify the denominator.

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Step 7.3.2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} - 1$

Step 7.3.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} - 1$

Step 7.3.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} - 1$

Step 7.3.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} - 1$

Step 7.3.2.5

Add $1$ and $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} - 1$

Step 7.3.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 7.3.2.6.2

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

$\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} - 1$

Step 7.3.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} - 1$

Step 7.3.2.6.4

Cancel the common factor of $2$ .

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Step 7.3.2.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} - 1$

Step 7.3.2.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{2}}{2^{1}} - 1$

Step 7.3.2.6.5

Evaluate the exponent.

$\frac{2 \sqrt{2}}{2} - 1$

Step 7.3.3

Cancel the common factor of $2$ .

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Step 7.3.3.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2} - 1$

Step 7.3.3.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2} - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2} - 1$

Decimal Form:

$0.41421356 \dots$