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Integrals

Long Answer Type

261.

Evaluate: $\int_{0}^{π} \frac{xsinx}{1 + \cos^{2} x} dx$

262.

Evaluate: $\int_{- a}^{a} \sqrt{\frac{a - x}{a + x}} dx$

Short Answer Type

263.

Evaluate: $\int \sec^{2} (7 - x) dx$

264.

If $\int_{0}^{1} (3 x^{2} + 2 x + k) dx = 0$ , find the value of k.

Long Answer Type

265.

Evaluate: $\int \frac{e^{x}}{\sqrt{5 - 4 e^{x} - e^{2 x}}} dx$

266.

Evaluate: $\int \frac{(x - 4) e^{x}}{{(x - 2)}^{3}} dx .$

267.
Evaluate: $\int_{0}^{π} \frac{e^{cosx}}{e^{cosx} + e^{- cosx}}$

$Let I = \int_{0}^{π} \frac{e^{cosx}}{e^{cosx} + e^{- cosx}} dx Using \int_{0}^{a} f (x) = \int_{0}^{a} f (a - x) dx I = \int_{0}^{π} \frac{e^{\cos (π - x)}}{e^{\cos (π - x)} + e^{- \cos (π - x)}} . dx 2 I = \int_{0}^{π} \frac{e^{- \cos x} + e^{\cos x}}{e^{\cos x} + e^{- \cos x}} . dx I = \frac{1}{2} \int_{0}^{π} dx = \frac{1}{2} [π - 0] I = \frac{π}{2}$