Calculus Examples

$\int_{0}^{\frac{π}{4}} \frac{1 + \cos^{2} (θ)}{\cos^{2} (θ)} d θ$

Step 1

Distribute denominator.

$\int_{0}^{\frac{π}{4}} \frac{1}{\cos^{2} (θ)} + \frac{\cos^{2} (θ)}{\cos^{2} (θ)} d θ$

Step 2

Simplify.

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Step 2.1

Convert from $\frac{1}{\cos^{2} (θ)}$ to $\sec^{2} (θ)$ .

$\int_{0}^{\frac{π}{4}} \sec^{2} (θ) + \frac{\cos^{2} (θ)}{\cos^{2} (θ)} d θ$

Step 2.2

Cancel the common factor of $\cos^{2} (θ)$ .

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Step 2.2.1

Cancel the common factor.

$\int_{0}^{\frac{π}{4}} \sec^{2} (θ) + \frac{\cos^{2} (θ)}{\cos^{2} (θ)} d θ$

Step 2.2.2

Rewrite the expression.

$\int_{0}^{\frac{π}{4}} \sec^{2} (θ) + 1 d θ$

Step 3

Split the single integral into multiple integrals.

$\int_{0}^{\frac{π}{4}} \sec^{2} (θ) d θ + \int_{0}^{\frac{π}{4}} d θ$

Step 4

Since the derivative of $\tan (θ)$ is $\sec^{2} (θ)$ , the integral of $\sec^{2} (θ)$ is $\tan (θ)$ .

$\tan (θ)]_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{4}} d θ$

Step 5

Apply the constant rule.

$\tan (θ)]_{0}^{\frac{π}{4}} + θ]_{0}^{\frac{π}{4}}$

Step 6

Simplify the answer.

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Step 6.1

Combine $\tan (θ)]_{0}^{\frac{π}{4}}$ and $θ]_{0}^{\frac{π}{4}}$ .

$\tan (θ) + θ]_{0}^{\frac{π}{4}}$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $\tan (θ) + θ$ at $\frac{π}{4}$ and at $0$ .

$(\tan (\frac{π}{4}) + \frac{π}{4}) - (\tan (0) + 0)$

Step 6.2.2

Add $\tan (0)$ and $0$ .

$\tan (\frac{π}{4}) + \frac{π}{4} - \tan (0)$

Step 6.3

Simplify.

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Step 6.3.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$1 + \frac{π}{4} - \tan (0)$

Step 6.3.2

The exact value of $\tan (0)$ is $0$ .

$1 + \frac{π}{4} - 0$

Step 6.3.3

Multiply $- 1$ by $0$ .

$1 + \frac{π}{4} + 0$

Step 6.3.4

Add $1 + \frac{π}{4}$ and $0$ .

$1 + \frac{π}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$1 + \frac{π}{4}$

Decimal Form:

$1.78539816 \dots$