What is the integral of 7x^6sin(x^7) ?

The integral of 7x^6sin(x^7) is -cos(x^7)+C

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integral of 7x^6sin(x^7)

\int 7 x^{6} sin (x^{7}) d x

- cos (x^{7}) + C

Solution steps

\int 7 x^{6} sin (x^{7}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 7 \cdot \int x^{6} sin (x^{7}) d x

Apply u-substitution

: \int \frac{sin ( u )}{7} d u

= 7 \cdot \int \frac{sin ( u )}{7} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 7 \cdot \frac{1}{7} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 7 \cdot \frac{1}{7} (- cos (u))

Substitute back

u = x^{7}

= 7 \cdot \frac{1}{7} (- cos (x^{7}))

Simplify

7 \cdot \frac{1}{7} (- cos (x^{7})) : - cos (x^{7})

= - cos (x^{7})

Add a constant to the solution

= - cos (x^{7}) + C

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\int_{- 3}^{3} \frac{14}{x ^{2} - 4 x - 45} d x

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What is the integral of 7x^6sin(x^7) ?
The integral of 7x^6sin(x^7) is -cos(x^7)+C