Calculus Examples

$\int_{0}^{1} 2 x^{2} \cos (x) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{1} x^{2} \cos (x) d x$

Step 2

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

$2 (x^{2} \sin (x)]_{0}^{1} - \int_{0}^{1} \sin (x) (2 x) d x)$

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 (x^{2} \sin (x)]_{0}^{1} - (2 \int_{0}^{1} \sin (x) (x) d x))$

Step 4

Multiply $2$ by $- 1$ .

$2 (x^{2} \sin (x)]_{0}^{1} - 2 \int_{0}^{1} \sin (x) (x) d x)$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (x)$ .

$2 (x^{2} \sin (x)]_{0}^{1} - 2 (x (- \cos (x))]_{0}^{1} - \int_{0}^{1} - \cos (x) d x))$

Step 6

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

$2 (x^{2} \sin (x)]_{0}^{1} - 2 (x (- \cos (x))]_{0}^{1} - - \int_{0}^{1} \cos (x) d x))$

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $- 1$ .

$2 (x^{2} \sin (x)]_{0}^{1} - 2 (x (- \cos (x))]_{0}^{1} + 1 \int_{0}^{1} \cos (x) d x))$

Step 7.2

Multiply $\int_{0}^{1} \cos (x) d x$ by $1$ .

$2 (x^{2} \sin (x)]_{0}^{1} - 2 (x (- \cos (x))]_{0}^{1} + \int_{0}^{1} \cos (x) d x))$

Step 8

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$2 (x^{2} \sin (x)]_{0}^{1} - 2 (x (- \cos (x))]_{0}^{1} + \sin (x)]_{0}^{1}))$

Step 9

Simplify the answer.

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Step 9.1

Combine $x (- \cos (x))]_{0}^{1}$ and $\sin (x)]_{0}^{1}$ .

$2 (x^{2} \sin (x)]_{0}^{1} - 2 (x (- \cos (x)) + \sin (x)]_{0}^{1}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $x^{2} \sin (x)$ at $1$ and at $0$ .

$2 ((1^{2} \sin (1)) - 0^{2} \sin (0) - 2 (x (- \cos (x)) + \sin (x)]_{0}^{1}))$

Step 9.2.2

Evaluate $x (- \cos (x)) + \sin (x)$ at $1$ and at $0$ .

$2 ((1^{2} \sin (1)) - 0^{2} \sin (0) - 2 ((1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3

Simplify.

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Step 9.2.3.1

One to any power is one.

$2 (1 \sin (1) - 0^{2} \sin (0) - 2 ((1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3.2

Multiply $\sin (1)$ by $1$ .

$2 (\sin (1) - 0^{2} \sin (0) - 2 ((1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3.3

Raising $0$ to any positive power yields $0$ .

$2 (\sin (1) - 0 \sin (0) - 2 ((1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3.4

Multiply $- 1$ by $0$ .

$2 (\sin (1) + 0 \sin (0) - 2 ((1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3.5

Multiply $0$ by $\sin (0)$ .

$2 (\sin (1) + 0 - 2 ((1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3.6

Add $\sin (1)$ and $0$ .

$2 (\sin (1) - 2 ((1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3.7

Multiply $- 1$ by $1$ .

$2 (\sin (1) - 2 (- \cos (1) + \sin (1) - (0 (- \cos (0)) + \sin (0))))$

Step 9.2.3.8

Multiply $- 1$ by $0$ .

$2 (\sin (1) - 2 (- \cos (1) + \sin (1) - (0 \cos (0) + \sin (0))))$

Step 9.2.3.9

Multiply $0$ by $\cos (0)$ .

$2 (\sin (1) - 2 (- \cos (1) + \sin (1) - (0 + \sin (0))))$

Step 9.2.3.10

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

$2 (\sin (1) - 2 (- \cos (1) + \sin (1) - \sin (0)))$

Step 9.3

Simplify.

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Step 9.3.1

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

$2 (\sin (1) - 2 (- \cos (1) + \sin (1) - 0))$

Step 9.3.2

Multiply $- 1$ by $0$ .

$2 (\sin (1) - 2 (- \cos (1) + \sin (1) + 0))$

Step 9.3.3

Add $- \cos (1) + \sin (1)$ and $0$ .

$2 (\sin (1) - 2 (- \cos (1) + \sin (1)))$

Step 9.4

Simplify.

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Step 9.4.1

Evaluate $\cos (1)$ .

$2 (\sin (1) - 2 (- 1 \cdot 0.5403023 + \sin (1)))$

Step 9.4.2

Multiply $- 1$ by $0.5403023$ .

$2 (\sin (1) - 2 (- 0.5403023 + \sin (1)))$

Step 9.4.3

Evaluate $\sin (1)$ .

$2 (\sin (1) - 2 (- 0.5403023 + 0.84147098))$

Step 9.4.4

Add $- 0.5403023$ and $0.84147098$ .

$2 (\sin (1) - 2 \cdot 0.30116867)$

Step 9.4.5

Multiply $- 2$ by $0.30116867$ .

$2 (\sin (1) - 0.60233735)$

Step 9.4.6

Evaluate $\sin (1)$ .

$2 (0.84147098 - 0.60233735)$

Step 9.4.7

Subtract $0.60233735$ from $0.84147098$ .

$2 \cdot 0.23913362$

Step 9.4.8

Multiply $2$ by $0.23913362$ .

$0.47826725$