Math / Calculus

QuestionFind $d u$ and $u$ \begin{tabular}{|l|} \hline 6. $\int \operatorname{Arctan} x d x$ \\ \hline 7. $\int \frac{\operatorname{Arcsin} 2 x d x}{\sqrt{1-4 x^{2}}}$ \\ \hline 8. $\int x \csc \left(1-x^{2}\right) \cot \left(1-x^{2}\right) d x$ \\ \hline 9. $\int 10^{\cos x} e^{\cos x} \sin x d x$ \\ \hline $10 \int x^{3} e^{2 x} d x$ \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. We are given a list of integrals to solve.
2. We need to find $du$ and $u$ for each integral.
3. We will use techniques such as substitution and integration by parts where applicable.

STEP 2

1. Solve integral 6: $\int \operatorname{Arctan} x \, dx$ .
2. Solve integral 7: $\int \frac{\operatorname{Arcsin} 2x \, dx}{\sqrt{1-4x^2}}$ .
3. Solve integral 8: $\int x \csc(1-x^2) \cot(1-x^2) \, dx$ .
4. Solve integral 9: $\int 10^{\cos x} e^{\cos x} \sin x \, dx$ .
5. Solve integral 10: $\int x^3 e^{2x} \, dx$ .

STEP 3

For $\int \operatorname{Arctan} x \, dx$ , use integration by parts: Let $u = \operatorname{Arctan} x$ and $dv = dx$ .
Then, $du = \frac{1}{1+x^2} \, dx$ and $v = x$ .

STEP 4

For $\int \frac{\operatorname{Arcsin} 2x \, dx}{\sqrt{1-4x^2}}$ , use substitution: Let $u = \operatorname{Arcsin} 2x$ .
Then, $du = \frac{2 \, dx}{\sqrt{1-4x^2}}$ .

STEP 5

For $\int x \csc(1-x^2) \cot(1-x^2) \, dx$ , use substitution: Let $u = 1-x^2$ .
Then, $du = -2x \, dx$ .

STEP 6

For $\int 10^{\cos x} e^{\cos x} \sin x \, dx$ , use substitution: Let $u = \cos x$ .
Then, $du = -\sin x \, dx$ .

STEP 7

For $\int x^3 e^{2x} \, dx$ , use integration by parts: Let $u = x^3$ and $dv = e^{2x} \, dx$ .
Then, $du = 3x^2 \, dx$ and $v = \frac{1}{2}e^{2x}$ .

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Studdy Solution

STEP 1

1. We are given a list of integrals to solve.
2. We need to find $du$ and $u$ for each integral.
3. We will use techniques such as substitution and integration by parts where applicable.

STEP 2

STEP 3

For $\int \operatorname{Arctan} x \, dx$ , use integration by parts: Let $u = \operatorname{Arctan} x$ and $dv = dx$ .
Then, $du = \frac{1}{1+x^2} \, dx$ and $v = x$ .

STEP 4

For $\int \frac{\operatorname{Arcsin} 2x \, dx}{\sqrt{1-4x^2}}$ , use substitution: Let $u = \operatorname{Arcsin} 2x$ .
Then, $du = \frac{2 \, dx}{\sqrt{1-4x^2}}$ .

STEP 5

For $\int x \csc(1-x^2) \cot(1-x^2) \, dx$ , use substitution: Let $u = 1-x^2$ .
Then, $du = -2x \, dx$ .

STEP 6

For $\int 10^{\cos x} e^{\cos x} \sin x \, dx$ , use substitution: Let $u = \cos x$ .
Then, $du = -\sin x \, dx$ .

STEP 7

For $\int x^3 e^{2x} \, dx$ , use integration by parts: Let $u = x^3$ and $dv = e^{2x} \, dx$ .
Then, $du = 3x^2 \, dx$ and $v = \frac{1}{2}e^{2x}$ .

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Studdy Solution

STEP 1

1. We are given a list of integrals to solve.2. We need to find du du du and u u u for each integral.3. We will use techniques such as substitution and integration by parts where applicable.

STEP 2

STEP 3

For ∫Arctan⁡x dx\int \operatorname{Arctan} x \, dx∫Arctanxdx, use integration by parts: Let u=Arctan⁡x u = \operatorname{Arctan} x u=Arctanx and dv=dx dv = dx dv=dx.Then, du=11+x2 dx du = \frac{1}{1+x^2} \, dx du=1+x21​dx and v=x v = x v=x.

STEP 4

For ∫Arcsin⁡2x dx1−4x2\int \frac{\operatorname{Arcsin} 2x \, dx}{\sqrt{1-4x^2}}∫1−4x2​Arcsin2xdx​, use substitution: Let u=Arcsin⁡2x u = \operatorname{Arcsin} 2x u=Arcsin2x.Then, du=2 dx1−4x2 du = \frac{2 \, dx}{\sqrt{1-4x^2}} du=1−4x2​2dx​.

STEP 5

For ∫xcsc⁡(1−x2)cot⁡(1−x2) dx\int x \csc(1-x^2) \cot(1-x^2) \, dx∫xcsc(1−x2)cot(1−x2)dx, use substitution: Let u=1−x2 u = 1-x^2 u=1−x2.Then, du=−2x dx du = -2x \, dx du=−2xdx.

STEP 6

For ∫10cos⁡xecos⁡xsin⁡x dx\int 10^{\cos x} e^{\cos x} \sin x \, dx∫10cosxecosxsinxdx, use substitution: Let u=cos⁡x u = \cos x u=cosx.Then, du=−sin⁡x dx du = -\sin x \, dx du=−sinxdx.

STEP 7

For ∫x3e2x dx\int x^3 e^{2x} \, dx∫x3e2xdx, use integration by parts: Let u=x3 u = x^3 u=x3 and dv=e2x dx dv = e^{2x} \, dx dv=e2xdx.Then, du=3x2 dx du = 3x^2 \, dx du=3x2dx and v=12e2x v = \frac{1}{2}e^{2x} v=21​e2x.

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1. We are given a list of integrals to solve.
2. We need to find $du$ and $u$ for each integral.
3. We will use techniques such as substitution and integration by parts where applicable.

For $\int \operatorname{Arctan} x \, dx$ , use integration by parts: Let $u = \operatorname{Arctan} x$ and $dv = dx$ .
Then, $du = \frac{1}{1+x^2} \, dx$ and $v = x$ .

For $\int \frac{\operatorname{Arcsin} 2x \, dx}{\sqrt{1-4x^2}}$ , use substitution: Let $u = \operatorname{Arcsin} 2x$ .
Then, $du = \frac{2 \, dx}{\sqrt{1-4x^2}}$ .

For $\int x \csc(1-x^2) \cot(1-x^2) \, dx$ , use substitution: Let $u = 1-x^2$ .
Then, $du = -2x \, dx$ .

For $\int 10^{\cos x} e^{\cos x} \sin x \, dx$ , use substitution: Let $u = \cos x$ .
Then, $du = -\sin x \, dx$ .

For $\int x^3 e^{2x} \, dx$ , use integration by parts: Let $u = x^3$ and $dv = e^{2x} \, dx$ .
Then, $du = 3x^2 \, dx$ and $v = \frac{1}{2}e^{2x}$ .