Calculus Examples

$x \tan^{2} (x)$

Step 1

Write $x \tan^{2} (x)$ as a function.

$f (x) = x \tan^{2} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x \tan^{2} (x) d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \tan^{2} (x)$ .

$x (- x + \tan (x)) - \int - x + \tan (x) d x$

Step 5

Split the single integral into multiple integrals.

$x (- x + \tan (x)) - (\int - x d x + \int \tan (x) d x)$

Step 6

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

$x (- x + \tan (x)) - (- \int x d x + \int \tan (x) d x)$

Step 7

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$x (- x + \tan (x)) - (- (\frac{1}{2} x^{2} + C) + \int \tan (x) d x)$

Step 8

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$x (- x + \tan (x)) - (- (\frac{1}{2} x^{2} + C) + \ln (| \sec (x) |) + C)$

Step 9

Simplify.

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Step 9.1

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

$x (- x + \tan (x)) - (- (\frac{x^{2}}{2} + C) + \ln (| \sec (x) |) + C)$

Step 9.2

Simplify.

$- x \cdot x + x \tan (x) - (- \frac{x^{2}}{2} + \ln (| \sec (x) |)) + C$

Step 9.3

Simplify.

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Step 9.3.1

Raise $x$ to the power of $1$ .

$- (x^{1} x) + x \tan (x) - (- \frac{x^{2}}{2} + \ln (| \sec (x) |)) + C$

Step 9.3.2

Raise $x$ to the power of $1$ .

$- (x^{1} x^{1}) + x \tan (x) - (- \frac{x^{2}}{2} + \ln (| \sec (x) |)) + C$

Step 9.3.3

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

$- x^{1 + 1} + x \tan (x) - (- \frac{x^{2}}{2} + \ln (| \sec (x) |)) + C$

Step 9.3.4

Add $1$ and $1$ .

$- x^{2} + x \tan (x) - (- \frac{x^{2}}{2} + \ln (| \sec (x) |)) + C$

Step 9.4

Simplify.

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Step 9.4.1

Apply the distributive property.

$- x^{2} + x \tan (x) - - \frac{x^{2}}{2} - \ln (| \sec (x) |) + C$

Step 9.4.2

Multiply $- - \frac{x^{2}}{2}$ .

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Step 9.4.2.1

Multiply $- 1$ by $- 1$ .

$- x^{2} + x \tan (x) + 1 \frac{x^{2}}{2} - \ln (| \sec (x) |) + C$

Step 9.4.2.2

Multiply $\frac{x^{2}}{2}$ by $1$ .

$- x^{2} + x \tan (x) + \frac{x^{2}}{2} - \ln (| \sec (x) |) + C$

Step 9.5

Reorder terms.

$- x^{2} + x \tan (x) + \frac{1}{2} x^{2} - \ln (| \sec (x) |) + C$

Step 10

The answer is the antiderivative of the function $f (x) = x \tan^{2} (x)$ .

$F (x) =$ $- x^{2} + x \tan (x) + \frac{1}{2} x^{2} - \ln (| \sec (x) |) + C$