Calculus Examples

$\int_{π}^{\frac{3 π}{4}} 7 \sec^{2} (t) d t$

Step 1

Since $7$ is constant with respect to $t$ , move $7$ out of the integral.

$7 \int_{π}^{\frac{3 π}{4}} \sec^{2} (t) d t$

Step 2

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

$7 (\tan (t)]_{π}^{\frac{3 π}{4}})$

Step 3

Evaluate $\tan (t)$ at $\frac{3 π}{4}$ and at $π$ .

$7 (\tan (\frac{3 π}{4}) - \tan (π))$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$7 (- \tan (\frac{π}{4}) - \tan (π))$

Step 4.1.2

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

$7 (- 1 \cdot 1 - \tan (π))$

Step 4.1.3

Multiply $- 1$ by $1$ .

$7 (- 1 - \tan (π))$

Step 4.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$7 (- 1 - - \tan (0))$

Step 4.1.5

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

$7 (- 1 - - 0)$

Step 4.1.6

Multiply $- - 0$ .

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Step 4.1.6.1

Multiply $- 1$ by $0$ .

$7 (- 1 - 0)$

Step 4.1.6.2

Multiply $- 1$ by $0$ .

$7 (- 1 + 0)$

Step 4.2

Add $- 1$ and $0$ .

$7 \cdot - 1$

Step 4.3

Multiply $7$ by $- 1$ .

$- 7$